matej.tekavcic@ijs.si

Three-dimensional simulations of liquid waves in isothermal vertical churn flow with OpenFOAM

Matej Tekavčič (), Boštjan Končar, Ivo Kljenak

Reactor Engineering Division, Jožef Stefan Institute, Jamova cesta 39, SI-1000 Ljubljana, Slovenia Abstract Periodic liquid waves of large amplitude are one of characteristic phenomena observed in the

churn flow regime of gas–liquid flow in vertical conduits, where the liquid flowing on the wall is entrained upwards by the gas flow in the core. The present work investigates the frequency of these large liquid waves. Three-dimensional simulations of isothermal churn flow of air and

water in 19 and 32 mm vertical pipes were performed using the interFoam solver from the OpenFOAM library. Turbulent features in the flow are modelled using the unsteady Reynolds Averaged Navier–Stokes approach with the k–ω SST (shear stress transport) model. Interface

sharpening with bounded compression was used to preserve the sharpness of gas–liquid interface by compensating the diffusive fluxes of the numerical scheme. A sensitivity study on the global amount of interface compression was performed for two flow cases taken from the literature.

Mesh sensitivity study was performed using four meshes, ranging from a coarse mesh with several hundred thousand cells, to a fine mesh with several million computational cells. Results for the calculated wave frequency and wave maximum amplitude agree with measured values reported

in the literature.

Keywords two-phase flow simulation

vertical churn flow

wave frequency

interface compression

Article History Received: 6 March 2019

Revised: 25 April 2019

Accepted: 26 April 2019

Research Article © Tsinghua University Press 2019

1 Introduction

The physical mechanisms that govern the formation and development of periodic large liquid waves in the churn flow regime can be related to the onset of the flooding phenomena (or the counter-current flow limitation, CCFL) in vertical conduits (Govan et al., 1991), which can occur during a hypothetical loss-of-coolant accident (LOCA) in light water nuclear reactors. Prediction of the onset of flooding in realistic geometries is still very uncertain (Vierow, 2008) and a more thorough understanding of the triggering mechanisms is required. Supported by reliable experiments with data measured on the local scales, simulations with computational fluid dynamics (CFD) approach can help us to discover the fundamental nature of such complex two-phase flows in light water reactors (Bestion, 2014).

Figure 1 shows an example of periodic wave formation in a vertical tube and showcases the region around a perforated liquid injector typically used in the experiments (Wang et al., 2013). Transparent injectors are used to record the features of the two-phase flow with a high-speed camera (for example 2000 frames per second). With post-processing

of visualization data, wave characteristics such as amplitude, frequency, velocity and shape can be obtained. Wave amplitude is defined as the distance between the pipe wall and the crest of the wave. Wave frequency is determined from the number of waves occurring over the duration of high-speed camera recording.

Three-dimensional simulations of liquid waves in isothermal churn flow regime of air and water in a vertical pipe were considered in the present paper. Two-phase flow was simulated with the single-fluid modelling approach based on the Volume-of-Fluid (VOF) method of Hirt and Nichols (1981) using the interFoam program from the open source library OpenFOAM (OpenCFD Ltd., 2015). The effects of turbulence in the flow were modelled using the Unsteady Reynolds Averaged Navier–Stokes (URANS) approach with the k–ω SST (shear–stress transport) turbulence model. The interface between gas and liquid was simulated using the interface capturing method with bounded compression (Rusche, 2002).

The computational domain in CFD simulation is divided into many computational cells with the finite volume method. In the single-fluid modelling approach with interface

Vol. 1, No. 4, 2019, 300–306Experimental and Computational Multiphase Flow https://doi.org/10.1007/s42757-019-0029-7

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capturing method, the interface is then captured on this fixed grid of computational cells using the volume fraction value of gas or liquid phase as the indicator function. The two phases are treated as an effective fluid mixture with variable properties according to whether the cell represents a region with gas, liquid, or the interface. For example, in our case the computational cell is either in the liquid wave or liquid film next to the pipe wall, or in the gas flow in the core of the pipe. In the transition region from the liquid wave to the gas flow, there are cells that include both phases (with volume fraction values between zero and one) and can represent the large interface of a liquid wave.

The phase boundary is represented by a steep gradient of the volume fraction. Solving the associated transport equation introduces an artificial (numerical) diffusion of the interface. Thus, the representation of the interface can be smeared over several computational cells in the normal direction to the surface. In addition, the phase volume fraction has to remain bounded between zero and one. Two groups of methods are used to tackle the described problem (Klostermann et al., 2012).

Reconstruction methods deal with the phase transport in two steps—geometric interface reconstruction and interface propagation. These methods represent the classic VOF approach, such as donor–acceptor method (Hirt and Nichols, 1981), simple line interface calculation (SLIC) (Noh and Woodward, 1976), and piecewise linear interface recon-struction (PLIC) (Youngs, 1982). Geometric reconstruction is difficult in three dimensions and on non-rectangular grids. Methods that can overcome this limit are computationally demanding (Klostermann et al., 2012).

On the other hand, the phase transport equation can be solved directly with high-resolution discretisation schemes

developed for this purpose (Klostermann et al., 2012). Among others, these include compressive interface capturing scheme for arbitrary meshes (Ubbink, 1997) and high-resolution interface-capturing scheme (Muzaferija and Peric, 1997).

The interface compression solution method used in the present work falls in the later category and was introduced in the CFD software OpenFOAM (OpenCFD Ltd., 2015) by Henry Weller, following the Ph.D. thesis of Rusche (2002). The method has been applied in interFoam solver and extensively analysed by Rusche (2002), Deshpande et al. (2012), and Klostermann et al. (2012). In their work, the methodology is named as surface compression (Klostermann et al., 2012) or bounded compression method (Rusche, 2002).

The intent of interface compression is to preserve the sharpness of gas–liquid interface by compensating the diffusive fluxes of the numerical scheme. To this end, an additional compression term is included to the transport equation of the volume fraction (see Eq. (4) in the following section), that is significant only in the computational cells which include the gas–liquid interface. The amount of interface compression in the model is controlled by a global parameter value ( αc in Eq. (11) in the following section), which is usually set to one.

Simulations (Tekavčič et al., 2018b) have shown that the formation and development of large liquid waves can be considered as a separate flow phenomenon. Here, keeping the interface sharp is required to properly resolve the shape of a large liquid wave. The performance of the interface compression method in interFoam has been analysed in the paper of Larsen et al. (2018) for the simulation of horizontal surface liquid waves. In the present work, the effect of the interface compression method on the simulated frequency of large liquid waves in vertical churn flow is studied. For

Fig. 1 Formation and migration of large liquid waves near the liquid inlet in the experiment of Wang et al. (2013) (wave visualizationadopted with permission via personal communication).

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this purpose, the amount of interface compression is varied from zero (no compression) to three.

2 Numerical simulation

Three-dimensional transient simulations were performed with the open source CFD package OpenFoam version 3.0.0 solver interFoam. A detailed description of the flow model of interFoam can be found in the open literature (Deshpande et al., 2012; Klostermann et al., 2012; Larsen et al., 2018). For the presented work, a numerical model from our previous simulations of liquid waves was adopted (Tekavčič et al., 2018b). A brief overview of the modelling approach is presented in this section.

2.1 Flow model

The volume fraction of water α was used as the phase indicator field which is defined as

1, in water0 1, at the interface0, in air

α αìïïïï= < <íïïïïî

(1)

The unsteady flow of incompressible homogeneous mixture is described by the continuity equation:

0⋅ =U (2)

The momentum (Navier–Stokes) equation:

*eff( ) σ

ρ ρ p ρt

¶+⋅ =- - ⋅ +⋅ +

¶

U UU g x F (3)

and the transport equation for the volume fraction α using the interface compression approach

) r( ( (1 ) ) 0α α α αt

¶+⋅ ⋅ - =

¶+U U (4)

The terms in Eqs. (2)–(4) are explained next. A single velocity field U for the effective fluid mixture

is used. The mixture density ρ and viscosity μ are averaged from the two constituent fluids (water: l, air: g):

( ) ( )l g l g1 , 1ρ αρ α ρ μ αμ α μ= + - = + - (5)

Modified pressure without hydrostatic contribution is used:

( )refp p ρ* = - ⋅ -g x x (6)

where g = (0, 0, g) is the gravitational acceleration downwards along the axis of the pipe, and x = (x, y, z) is the coordinate vector. The reference height was set to the outlet plane at the top of the domain at xref = (0, 0, 0).

The effective stress of the homogeneous mixture eff is

the combined viscous stress and turbulent stress Τ :

Τ Teff , ( )μ= + + U U (7)

The closure relation for the turbulent stress is obtained by the k–ω (shear–stress transport) turbulence model (Menter, 1994). This is a two-equation linear eddy viscosity model obtained with Unsteady Reynolds Averaged Navier– Stokes (URANS) approach.

The surface tension source is modelled using the continuum surface force model of Brackbill et al. (1992):

σ σκ α κ *= , =-⋅F n (8)

where κ is the surface curvature. The interface unit normal vector *n is calculated from the interpolated face-gradient of the volume fraction f( )α ：

( )

( )

8f

n 1 3n cf

10αδ

δα V

-*

/

= , =

+ n (9)

where nδ is a small number added for numerical stability, and cV is the average volume of computational cell in the mesh. A constant surface tension coefficient was prescribed with 0 072 N/mσ = . .

Equations (2) and (3) were solved using the PISO (pressure implicit with splitting of operator) coupling algorithm (Rusche, 2002; OpenCFD Ltd., 2015). The boundedness of 0 1α£ £ in Eq. (4) is maintained with the multidimensional universal limiter for explicit solution (MULES) algorithm (Deshpande et al., 2012).

The interface compression term r( (1 ) )α α⋅ - U in Eq. (4) is active only near the interface ( 0 1α< < ). The model for the relative velocity rU depends on the discre-tisation procedure of interFoam which is described in Deshpande et al. (2012), Klostermann et al. (2012), and Larsen et al. (2018). In brief, the interface compression term in Eq. (4) can be written in discrete form as Klostermann et al. (2012):

Ur r( (1 ) )d (1 )f fV f

α α V α α⋅ - = -åò (10)

where fα is the interpolated value on the face f of cell with cell volume V, with the summation over all cell faces. The compressive flux r over cell face f is calculated as

( )

r α ff

c *æ ö÷ç ÷= ⋅ç ÷ç ÷çè ø

n SS

(11)

where the interface normal *n is projected onto the cell face with vector fS . The face volumetric flux is defined with face interpolated velocity as f f= ⋅U S .

The user-specified parameter αc controls the amount of interface compression and should be in the range of

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unity (Deshpande et al., 2012; Klostermann et al., 2012). In the thesis of Rusche (2002), which can be considered as the origin of the method implemented in interFoam, the value

αc = 1.5 was used. In the present paper, the value αc is varied from zero (no compression) to three in order to quantify its effect on the frequency of large liquid waves.

2.2 Computational domain and boundary conditions

Simulations are based on the flow conditions shown in Table 1, that were taken from the two sets of air–water churn flow experiments with two different tube diameters. The computational domain, shown in Fig. 2, represents a region of a vertical pipe around the perforated wall inlet section for the liquid phase used in experiments (shown in Fig. 1). An overview of mesh average cell sizes calculated from the cell volumes reported by the OpenFOAM checkMesh tool is presented in Table 2. Simulations ran in parallel on a computer cluster using from 20 to 120 (on the mesh with 4 million cells) high performance Intel Xeon processor cores.

The upper and lower sections of the cylinder outer surface (Lw = 50 mm) were set as a no-slip wall boundary surface. The liquid inlet boundary with prescribed mass flow rate lm lies in-between the two wall sections. Tekavčič

Table 1 Overview of simulated flow conditions adopted from experiments. Properties of water were the same in both cases: ρl = 997.07 kg/m3 and νl = 1.004×10−6 m2/s

Ref. D (mm)

lm (g/s)

gj (m/s)

p0 (105 Pa)

ρg (kg/m3)

νg (m2/s)

Meas. f (Hz)

Wang et al. (2013) 19 25 5.0 1.00 1.177 1.568×10−5 10.4

Barbosa Jr. et al. (2001) 32 51 5.3 1.33 1.549 1.201×10−5 15.9

Fig. 2 (a) Cross-section and (b) side view of mesh cells near the liquid inlet section. Dimensions are not to scale.

Table 2 Mesh overview with characteristic cell sizes min max avgh / / = cell3

min max avgV / /

D (mm) H (mm) Ncells hmin (mm) hmax (mm) havg (mm)

coarse 131600 0.42 0.86 0.65 mid1 1052800 0.21 0.43 0.33 mid2 2105600 0.16 0.34 0.26

fine

19 30

4211200 0.13 0.27 0.21 coarse 32 12 330240 0.40 0.85 0.65

et al. (2018a) showed a strong correlation between the prescribed boundary liquid velocity (determined from the mass flow rate and inlet height) and the simulated wave frequency. Good agreement between the measured and simulated wave frequency was obtained with inlet height of H = 30 mm in 19 mm diameter pipe, and H = 12 mm in 32 mm diameter pipe, which were therefore also used in the present model. Pure gas flow was imposed at the gas inlet. Mapping was used for the boundary velocity field, which was sampled from a cross-sectional surface at the middle of the lower pipe section. To preserve the mass flow rate given by the superficial velocity gj , the area average of velocity was constrained. Similarly, turbulence fields k and ω at the gas inlet were mapped, but without constraints on their averages. The two-phase outlet is modelled as an opening boundary condition with modified pressure set to 0 Pap* = and zero-gradient condition applied to other fields.

3 Results and discussion

The validity of the presented modelling approach was confirmed by comparing the simulated results with measurements from the literature (Barbosa Jr. et al., 2001; Wang et al., 2013). Qualitatively, the occurrence and development of large liquid waves was examined using visualizations of the flow. The simulated phenomena exhibit the characteristic features of liquid waves in churn flow that were observed in the experiments. Figure 5 in Section 3.3 shows an example of instantaneous volume fraction field of water (α) at different simulation time on the radial–axial (r–z) plane cross-section of the pipe.

Quantitatively, simulated wave frequency was compared with the measured wave frequency, as explained in the next section. The sensitivity of the calculated wave frequency on the number of mesh is explored in Section 3.2. Finally, the study on the effects of interface compression is presented in Section 3.3. Results show a comparison between the measured and simulated wave frequency and maximum wave amplitude at different amounts of interface compression.

3.1 Simulated wave frequency

Similarly to the measured wave frequency in the experiments

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of Barbosa Jr. et al. (2001) and Wang et al. (2013), the simulated wave frequency was calculated from the number of observed waves N over the time period tD :

1Nft-

=D

(12)

Transient simulations were run with a total duration of 5 s. To exclude effects of initial conditions, counting of waves started after 1 s of simulation time, giving the wave observation time period of 4tD = s. Figure 3 shows peaks in the calculated pressure p* at the centre of the gas inlet that correspond to large liquid waves. The number of liquid waves was determined with an automated algorithm that detected the large oscillations in the recorded pressure.

To illustrate, for the case in 19 mm pipe with lower wave frequency of approximately 10 Hz, around 10 waves were observed in the initial 1 s of the simulation, and then around 40 waves were counted in the subsequent 4 s of the simulation. Furthermore, approximately 150 flow through times were simulated in 4 s for the case in 19 mm pipe with gas velocity of 5 m/s and domain length of 0.13 m.

Results in Fig. 3 show, that the periodic nature of liquid waves could not be reproduced on the coarse mesh without interface compression and a steady pressure signal was obtained. The effects of interface compression on the simulated wave frequency are presented in greater detail in Section 3.3.

3.2 Mesh sensitivity

Mesh sensitivity study was performed for the case in 19 mm pipe using four mesh configurations presented in Table 2. Figure 4 compares the simulated and measured wave frequency (Wang et al., 2013) at different mesh densities. Here, results are plotted in terms of the relative change in

Fig. 3 Peaks in the calculated pressure p* at the centre of the gas inlet with small amount (cα = 0.25) and without interface compression (cα = 0) obtained using the coarse mesh (see Table 2) for the case in 19 mm pipe (see Table 1).

Fig. 4 Mesh sensitivity of the simulated wave frequency compared to the measured frequency (Wang et al., 2013) for the case in 19 mm pipe (see Table 1) using interface compression with cα = 2. Cell sizes h are plotted relative to the cell size of the coarse mesh hcoarse.

cell size coarseh h/ compared to the cell size of the coarse mesh. Results show around 15% variation of the simulated wave frequency with decreasing size of computational cells. This is still an acceptable accuracy in terms of the overall complexity of considered multiphase flow phenomena.

3.3 Interface compression level

A sensitivity study on the amount of interface compression in the model (cα) was performed for cases in 19 and 32 mm pipes (see Table 1). Two representative meshes were used for the case in 19 mm pipe: the coarse grid with around hundred thousand cells and the fine grid with around four million computational cells (see Table 2). Only the coarse mesh was used for the case in 32 mm pipe. As already shown in Fig. 3, the simulation model using the coarse mesh and without interface compression was not able to predict the periodic nature of liquid waves. Conversely, on the fine mesh periodic waves were obtained also without interface compression.

Figure 5 shows the simulated water volume fraction field using the fine mesh for the case in 19 mm pipe at three different simulation time (in three rows) and four levels of interface compression (in four columns). Shown simulation time is separated by around one wave cycle (approx. 0.1 s at 10 Hz). Comparing the results with and without interface compression, there is more numerical diffusion of the interface in the later case, which affects the wave shape and to a lesser extent the simulated wave frequency.

Figure 6 shows the effect of interface compression level on the simulated wave frequency. Comparison of results with or without interface compression is only possible on the fine mesh, which was used only for the case in 19 mm pipe. Results show that the wave frequency is approximately 15% higher for the simulation without interface compression, which is comparable to the mesh sensitivity of the model observed in Section 3.2. Otherwise a particular choice for

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Fig. 5 Water volume fraction at different interface compression levels cα using the fine mesh in 19 mm pipe.

Fig. 6 Ratio of simulated wave frequency and measured frequency for the case in 19 mm pipe (Wang et al., 2013) and 32 mm pipe (Barbosa Jr. et al., 2001) with different interface compression levels cα on the coarse and fine meshes (see Table 2).

the interface compression level (cα) has no significant effect on the fine mesh.

Similar observation can be made with results on the coarse mesh for the two flow cases in 19 and 32 mm pipes. There is around 15% variation of the simulated frequency in 19 mm pipe and –10% in 32 mm pipe with interface compression level below one (and above zero). Above cα = 1, the interface compression has no significant effect on the simulated frequencies in both flow cases.

Another quantitative assessment of results can be made with the wave maximum amplitude, that was measured in the experiment of Wang et al. (2013) in 19 mm pipe. In the simulation, the wave maximum amplitude can be conveniently observed in the time averaged volume fraction field of water ( α ). Here, the volume fraction field was averaged over the 4 s simulation time, the same period that was used to determine the wave frequency. Figure 7 shows the time averaged field α obtained with different interface compression levels. The vertical blue line shows the wave maximum amplitude measured in the experiments (Wang et al., 2013). Present results show good agreement between the simulated and measured maximum wave amplitude with interface compression level of one or above. With interface compression level below one (including the case without interface com-pression), simulations predict waves with slightly smaller maximum amplitude.

4 Conclusions

Presented simulations of vertical churn flow of air and water with the interface compression model exhibit the characteristic features of large liquid waves, that qualitatively agree with the phenomena observed in experiments. The presented results show that the periodic nature of liquid waves could not be reproduced without interface compression, if the mesh is not adequately refined, for example on the coarse mesh in the present simulation. On the other hand,

Fig. 7 Averaged water volume fraction with different interface compression levels cα using the fine mesh for the case in 19 mm pipe (see Table 1).

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the small variations of wave frequency observed at different interface compression levels were comparable to the mesh sensitivity of the model and were within the limits of acceptable accuracy. Finally, good quantitative agreement between the simulated and measured frequency can be obtained in both examined flow cases (in 19 and 32 mm pipes) from the literature. Similarly, simulated maximum wave amplitudes agree with the measurement in 19 mm pipe.

Acknowledgements

The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P2-0026 “Reactor engineering”).

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