Direct numerical simulation of a turbulent bubbly flow

Nuclear Engineering and Design xxx (2017) xxx–xxx

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Direct numerical simulation of a turbulent bubbly flow in a vertical channel: Towards an improved second-order reynolds stress model Guillaume Bois Den-Service de thermo-hydraulique et de mécanique des fluides (STMF), CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

h i g h l i g h t s DNS of an upward turbulent bubbly flow in a plane channel is presented. Deformable bubbles are tracked using the Front-Tracking algorithm of TrioCFD. An up-scaling approach from DNS towards two-phase RANS CFD modelling is presented. Simulations of the averaged flow are performed within NEPTUNE_CFD. Turbulence models (SSG and EBRSM) are compared to the DNS reference data.

a r t i c l e

i n f o

Article history: Received 28 April 2016 Received in revised form 16 January 2017 Accepted 21 January 2017 Available online xxxx Keywords: Direct numerical simulation Bubbly flow Turbulence Reynolds stress model

a b s t r a c t Two-phase turbulence has been studied using a Direct Numerical Simulation (DNS) of an upward turbulent bubbly flow in a so-called plane channel. Fully deformable monodispersed bubbles are tracked by a Front-Tracking algorithm implemented in TrioCFD code on the TRUST platform. Realistic fluid properties are used to represent saturated steam and water in pressurised water reactor (PWR) conditions. The large number of bubbles creates a void fraction of 10%. The Reynolds friction number is 180. Time- and space-averaging is used to compute the main variables of the averaged scale description (e.g. void fraction, liquid and vapour velocities. . .) along with the Reynolds stresses and the turbulent dissipation rate tensor. Altogether, they provide reference profiles to assess and further improve Reynolds Stress models. A low-Reynolds version of the SSG model (Speziale et al., 1991) called EBRSM (Manceau and Hanjalic´, 2002; Manceau, 2005) is applied in the context of two-phase flows with additional interfacial production terms. The model has been implemented and tested in the two-fluid Euler-Euler model of NEPTUNE_CFD code. The comparison with DNS demonstrates that the interfacial momentum closure plays a dominant role over the turbulent closure hypothesis in the present physical conditions. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Two-phase bubbly flows are found in many engineering applications. They involve a wide range of scales, from the Kolmogorov scale to the macroscopic flow structures, and in between, the bubble diameter. For industrial applications such as Nuclear Reactor Safety analysis, it is essential to correctly model the main characteristics of such flows. Originally, onedimensional averaged models have been developed based on empirical correlations. Then, 3D-models that are averaged on finer space- and time-scales have been developed in the context of CMFD (Computational MultiFluid Dynamics) (Guelfi et al., 2007). In the quest of reduced uncertainties, the use of Reynolds Averaged

E-mail address: [email protected]

Navier Stokes (RANS) two-fluid equations is the most reliable approach, but the accuracy of the predicted results depends on the constitutive relations used to close the turbulent and interfacial transfers. Those models rely on local correlations very difficult to establish based on experiments. Hence, in this paper, we focus on the realisation of an up-scaling approach from local-scale simulations towards two-phase RANS CFD modelling. This approach aims at extracting information (such as correlations) from finescale simulations in order to suggest or calibrate new models for the Reynolds stresses or the interfacial momentum transfer. Direct Numerical Simulations (DNS) of two-phase flow being used as ‘‘numerical experiments” are then an excellent tool to develop local closures to the averaged models because they grant access to local quantities. In this paper, we present a first up-scaling step in which emphasis is laid on the turbulent fluxes, leaving aside the matter of interfacial transfers. The general

http://dx.doi.org/10.1016/j.nucengdes.2017.01.023 0029-5493/Ó 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: Bois, G. Direct numerical simulation of a turbulent bubbly flow in a vertical channel: Towards an improved second-order reynolds stress model. Nucl. Eng. Des. (2017), http://dx.doi.org/10.1016/j.nucengdes.2017.01.023

2

G. Bois / Nuclear Engineering and Design xxx (2017) xxx–xxx

Table 1 Review of turbulent bubbly flows simulations: main simulation parameters. N. A.: Not Available. Lengths are given either in wall unit defined from the Reynolds friction number (1w:u: ¼ h=Res , also called viscous unit) or in dimensionless unit 1d:u: ¼ h based on the channel half-width h. The inclination of the channel can be horizontal or vertical (upward or downward). When g ¼ 0 has been reported, it indicates that the computation does not take gravity into account. nb is the total number of bubbles tracked in the computational domain. Author

Kanai and Miyata (2001)

Kawamura and Kodama (2002)

Domain size (d:u:) Resolution

1 1 1 64 64 64 2.81 2.81 Air/water Horiz. 180 28.8 0.16 10.2 27 6%

6.4 2 3.2 64 64 64 N. A. N. A. Air/water Horiz. 180 71–113 0.4–0.63 4–12 9–54 3–8.6%

Dy (w:u:) min/max Dx; Dz (w:u:) Fluids Inclination Res Db (w:u:) Db (d:u:) Db =D nb

a

Lu et al. (2005)

Lu et al. (2006)

Lu and Tryggvason (2008)

Lee et al. (2014)

Bois G. 2p 2

p

p

p

p or 2p

2

2

2

2

p=2

p=2

p=2

p=2 or p

p

256 256 128 0.21/4.67 2.20 Air/water g¼0 180 54 0.4 11–326 16 5.4%

192 160 96 0.79/2.19 2.08 Air/water Downward 127.3 31.8 0.25 14–40 18–72 1.5–6.0%

256 192 128 0.21/4.67 1.56 Air/water Upward 127.2 38.3 0.30 8–231 21 3.0%

256 or 512 192 or 192 128 or 256 0.21/4.67 or 0.42/9.3 1.56 Air/water Upward 127–280 38 or 19 0.3 8–231 21 or 84 3.0%

384 1152 192 0.3125 2.94 Liq./vap. Upward 180 36 0.2 12–115 942 10%

interest for industrial applications covers a wide range of very different flows, from classical single-phase turbulent flows, to very complex boiling flows (with many different topological regimes). As a first step away from single-phase turbulence, we focus on an adiabatic bubbly flow, between two infinite parallel walls. This article starts with a brief overview of existing DNS of turbulent bubbly flows (Section 2). Then, the characteristics of the test-case and the numerical method are described in Section 3. The twofluid model is presented in Section 4 and applied to the DNS configuration in Section 5. Parametric studies on turbulence modelling are performed. Finally, conclusions and prospects are drawn in Section 6.

2. Overview: DNS of turbulent bubbly flows The first DNS of single-phase flow between two parallel walls has been performed by Kim et al. (1987), for a friction Reynolds Number Res ¼ qus h=l of 180, where q is the density, h is the channel half-width, l is the liquid viscosity and us is the wall friction pffiffiffiffiffiffiffiffiffiffiffi velocity defined by us ¼ sw =q, where sw is the mean shear stress at the wall. Increasing computational power has enabled the rise of the Reynolds number to the value of Res ¼ 2003 (Hoyas and Jimenez, 2006; Hoyas and Jimenez, 2008) and very recently Res ¼ 5200 has been simulated (Lee et al., 2014). DNS of turbulent two-phase flows are much more recent but it has already proven very useful to better understand the influence of non-dimensional parameters on the flow structure. A comprehensive review of DNS of bubbly flows is presented in Tryggvason et al. (2006), Tryggvason et al. (2013). Because of the increased complexity of two-phase flow compared to the standard single-phase flow, the first simulations focused on laminar (or pseudo-turbulent) flows. The first DNS of a turbulent bubbly channel flow with explicit tracking of deformable bubbles has been performed by Kanai and Miyata (2001). A few studies on bubbles and turbulence interaction have followed (e. g., Kawamura and Kodama, 2002; Lu et al., 2005; Lu et al., 2006). In particular, the interest in fully-resolving the bubbles’ deformations has been stated by Tryggvason et al. (2006) who demonstrated that the bubbles deformation strongly affects the drag coefficient and the lift force, hence resulting in very different void fraction profiles depending on the value of the Eötvös number Eo ¼ qgD2b =r where g is the acceleration due to

gravity, Db is the bubble diameter and r is the surface tension. Besides, the effect of the direction of gravity has been studied on air/water upward and downward flows (Lu et al., 2006; Lu and Tryggvason, 2008). More recently, Dabiri and Tryggvason (2015) have moved towards the study of convective heat transfer in turbulent bubbly up-flows. They studied the effect of multiphase fluid dynamics on heat transfers, neglecting phase-change and coalescence. The simulated flows reached a Reynolds number of Res ¼ 280, with up to 84 bubbles. Lately, Tryggvason and Lu (2015) have also shown interest in bubbles of different sizes rising upward in turbulent channel flow. The main characteristics of some of those studies are summarised in Table 1. To our knowledge, no DNS of high-pressure steam-water turbulent bubbly flow has been achieved yet. The present study is a novelty because we have simulated an upward bubbly flow with 10% void fraction in Pressurised Water Reactor (PWR) conditions, for a friction Reynolds number of 180. The void fraction of 10% is a great improvement compared to the existing literature, even though the achievable Reynolds number is still too low compared to most industrial applications. 3. Numerical method and computational setup Here, we simulate the rise of buoyant bubbles in turbulent upflow for pressurized steam/water conditions. We first describe the governing equations and the Front-Tracking method used to simulate the flow (Section 3.1). Then, the test-case and the computational domain are described (Section 3.2). Finally, elements of validation on single-phase turbulent flow are given (Section 3.3). 3.1. Governing equations and numerical method A finite-difference method with Front-Tracking is used to perform the numerical simulations. The ‘‘one-fluid” Navier–Stokes equation (Kataoka, 1986; Bunner and Tryggvason, 2003)

h i @ qu þ r ðqu uÞ ¼ rP þ qg þ r l ru þ rT u þ rjnv di @t ð1Þ is solved over the whole domain, including both the bubbles and the liquid. Here, u is the velocity vector, P is the pressure, q and l are the discontinuous density and viscosity fields respectively



(assumed constant for each phase), g is the gravity vector,

r is the

constant surface tension, di is a three-dimensional delta function located over the interfaces, j ¼ rs nv is twice the mean curvature (usually negative for bubbles) and nv is the unit vector normal to the interface, orientated towards the liquid. The last term on the RHS of Eq. (1) ensures that the momentum equation implicitly contains the correct stress boundary condition at the interface. Both phases are considered incompressible, so the mass conservation equation for the whole flow field reduces to:

ru¼0

ð2Þ

As we consider incompressible phases with constant viscosity, the diffusive effect of rT u is limited to an interfacial contribution: rT u rl. We have performed complementary tests (not shown here) for which no significant effect of this term has been observed in the case of bubbly flows with the characteristics studied here. Therefore, this term is not considered in the DNS presented in this paper. In order to avoid spurious current, the Navier–Stokes Eq. (1) can be formulated as follows

@ qu þ r ðqu uÞ ¼ rPnum þ r ðlruÞ þ Si þ S f @t

ð3Þ

where the source term S f ¼ Sfx ex is introduced in the streamwise direction to control the flow rate and to balance, on average, the Archimedean thrust and the wall shear stress D E Sxf ¼ vv Dqg þ sw0 =h with Dq ¼ ql qv . The angle brackets hi indicates time- and space-averaging over the whole computational domain (xyz and t). The surface tension term is computed along with the gravitational and possible repellent forces:

Si ¼ ðrj Dqg xr /r Þrvl

ð4Þ 1

Here, xr is the position vector of the real interface and /r is a repellent potential (detailed in Section 3.2). The averaged pressure gradient in the channel generated by gravity and wall friction is modelled by Sf and we resolve a periodic numerical pressure Pnum defined as:

Pnum ¼P ^ x qg þ

sw h

ð5Þ

The Navier–Stokes equations are solved by a projection method (Puckett et al., 1997), using fourth-order central differentiation for the evaluation of convective and diffusive terms on a fixed, staggered Cartesian grid. Fractional time stepping leads to a thirdorder accurate time advancing scheme (Low-storage Runge Kutta 3, Williamson, 1980). In the two-step prediction-correction algorithm, the source term Si is added to the main flow source term S f and to the evaluation of the convection and diffusion operators in order to obtain the predicted velocity. Then, an elliptic pressure equation is solved by an algebraic multigrid method to impose a divergence-free velocity field. This part of the algorithm is responsible for most of the computational power consumption (60–80% of the whole CPU time), hence specific efforts have been made to improve its efficiency. The new data structure, stored on a Cartesian grid structured by 3 indices ði; j; kÞ allows heavy strongscaling with approximately 150 000 elements per processor. Convection and diffusion operators have also been revised to avoid non-contiguous memory access and to improve the use of cache memory. 1 The marker’s position is corrected by Lx ¼ 2ph if it belongs to a virtual portion of a bubble in order to retrieve the correct Archimedean thrust over a bubble when it crosses the periodic boundary.

3

The algorithm is implemented in the TrioCFD code developed by CEA relying on the TRUST platform (formerly known as Trio_U, Calvin et al., 2002). The code is fully parallel, written in C++ and has been widely used for applications on single-phase fluid dynamics (e. g., Chandesris et al., 2013; Toutant and Bataille, 2013; Aulery et al., 2015). In addition, a Front-Tracking algorithm is implemented. The principle of the general method is very similar to the algorithm presented in Bunner and Tryggvason (2003). Some different choices have been made and are briefly presented in Mathieu (2004). Some major changes were, however, necessary for this simulation, such as higher order differencing schemes, periodic boundary conditions, repellent forces and main flow source term. A brief presentation of the key features of the method is presented hereafter. The interface (also called ‘‘the front”) is followed explicitly by connected marker points that form an unstructured triangular grid. The markers are advected by the velocity field interpolated from the fixed grid. The tangential component of the velocity in the bubble reference frame is responsible for progressive markers accumulation at the rear of the bubbles. In order to improve the mesh quality preservation and to limit the use of smoothing algorithm, the normal component alone is computed and used in the markers transport. As the front deforms, surface markers are dynamically added and deleted in order to preserve a constant accuracy in the interface description. The front is then used to compute the phase indicator function vl ¼ 1 in the liquid and 0 in the vapour phase, the density and the viscosity at each Eulerian grid point. A semi-local volume-preserving algorithm is used to ensure the conservation of both phases. Besides, Mathieu (2003) developed the original formulation (4) for the gravitational force that is discretized along with the surface tension force. This formulation releases the exact momentum conservation in favour of a numerical scheme free of any spurious currents. Mathieu (2003) proves that with the new numerical formulation, the mechanical energy of the discrete system decreases when it is isolated, hence allowing an equilibrium state with a zero velocity field.

3.2. Test-case description The computational domain is a rectangular channel bounded by two vertical walls (normal to y-axis) and periodic boundary conditions in the spanwise (z) and in the streamwise (x) directions. The channel width 2h is 1cm as described in Fig. 1. Contrary to previous works (e.g., Lu et al., 2005 and following applications), we did not use the so-called ‘‘minimum turbulent channel” (ph 2h ph=2) presented by Jiménez et al. (1991) because while turbulent flow could be sustained in that periodic box, we noticed significant discrepancies on the root mean square (rms) velocity fluctuations in the single-phase flow simulation compared to the reference results of Kim et al. (1987). Hence, we used a larger channel of 2ph 2h ph in the streamwise, spanwise and wall-normal directions, respectively. A uniform mesh with a finer resolution in the wall-normal direction was adopted so as to accurately capture the turbulent structures, while maintaining a satisfactory resolution of the bubbles and their deformations independently of their location in the channel. In this way, we eliminate potential competition between numerical and physical effects in the determination of the wall-normal bubble distribution. The numerical resolution is given in Table 1. The domain has been discretized with a uniform mesh of 85 million hexahedral cells, thus resolving the viscous sublayer at the wall up to yþ ¼ 0:3125 in the cell adjacent to the wall. No slip boundary condition has been used at the wall and periodic conditions were set in the other directions. Careful attention has been devoted to the consideration of physical properties of


4


added to the axial momentum equation. In actual fact, DNS performed on single-phase flows have shown the tendency to lead to a faster convergence of statistics when this source term is not constant in time and some fluctuations are allowed. This source represents the fluctuations of the mean pressure gradient over the computational domain dP=dx that cannot be resolved directly with periodic boundary conditions. At steady state, the averaged pressure gradient dP=dx and the weight of the mixture are balanced by shear stresses in the liquid (Lu and Tryggvason, 2008)

@s þ b þ ðqðyÞ hqiÞg ¼ 0 @y

ð6Þ

where s ¼ lðdu=dyÞ þ q u0 v 0 is the sum of the viscous and turbulent shear stresses. b ¼ dP=dx þ hqig is the time-averaged total pressure gradient, consisting in the sum of the imposed pressure gradient and the hydrostatic pressure gradient due to the weight of the mixture. qðyÞ is the density averaged over time and the ðxzÞ plane for a given y coordinate whereas hqi is the mean density taken over the whole domain. Hence, the last term on the LHS of Eq. (6) is zero if the mixture is completely homogeneous. The instantaneous source

Fig. 1. Description of the DNS set-up.

term Sxf leading to the expected time-averaged value of b evolves according to the following relaxation equation Table 2 Physical properties of saturated water and steam (T sat ¼ 618 K, Psat ¼ 15:5 MPa). Liquid

q l r

[kg m3] [Pa s] [J m2]

Vapour

594.38

101.93 6

6

23:108 10

68:327 10

4:6695 103

water and saturated steam similar to PWR conditions (under a pressure of 155bars). The physical properties are summarised in Table 2. The non-dimensional parameters describing the flow are the void fraction a ¼ 10%, the ratio of the bubble mean diameter over the channel half-width Db =h ¼ 0:2, the Atwood, the Eötvös, the Archimedes, the Morton and the Reynolds friction numbers given by:

At ¼ ðql qg Þ=ql ¼ 0:83;

Eo ¼ ðql qg ÞgD2b =r ¼ 1:03;

Ar ¼ Eo3=2 At1=2 Mo1=2 ¼ ql ðql qv ÞgD3b =l2l ¼ 6:15 105 ; Mo ¼ g l4l =ðql r3 Þ ¼ 3:53 1012 and Res ¼ ql us h=ll ¼ 180; The computation was initialised with a uniform distribution of 942 initially-spherical bubbles so as to occupy 10% of the total volume of the channel. The bubbles are fully deformable. The volumetric flow rate and the bubble volume are controlled. Thus, after a transitional regime, the flow reaches a statistical steadystate from which statistics can be computed. Converged statistics have been gathered over an averaging period of T ¼ 3:34 s of physical time which corresponds to 4 crossing of the channel (of length 2ph), or a non-dimensional period of T þ ¼ T sw =ll ¼ 498 viscous units. Since the flow variables are spatially homogeneous in x- and z-directions, the statistical average (noted with an overbar ) is obtained by averaging over time and over the ðxzÞ plane. Every timestep has been used to compute statistics for any cell-centre along the wall-normal axis (direction y). Because of flow symmetry, only the averages of the left and right-hand side of the channel are presented. Flow-rate control The motion of the bubbles under the effect of buoyancy forces drags the liquid upward (þx). The flow rate must be controlled in order to achieve the Reynolds friction number of Res ¼ 180 for which the mesh resolution has been designed. To do so, a time-dependant uniform volumetric source term S f ðtÞ is

@Sxf x0 pffiffiffiffiffi pffiffiffiffiffiffi ð s0 sw Þ ¼ @t h

ð7Þ

where x0 ¼ 500 s1 is the relaxation frequency and s0 ¼ 0:101kg. m1 s2 is the objective value of the wall friction leading to a Reynolds number of Res ¼ 180. sw is the instantaneous wall shear computed from the velocity field in the cells adjacent to the wall and averaged over both walls. Bubble-volume control In our simulation, following the path of previous works (e.g., Lu and Tryggvason, 2008; Dabiri and Tryggvason, 2015), we do not allow the bubbles to coalesce and the surface tension is sufficiently large to avoid bubbles’ breakup. This restriction is the consequence of some limitations of interface tracking methods that are not yet fully capable of accurately predicting coalescence and break-up at a reasonable cost. Even though the Front-Tracking method is a priori the most practical tool to implement subgrid models, to the best of the author’s knowledge, there is no relation yet able to predict coalescence based on physical principles or laws on local quantities. Hence, any local-scale simulation has a tendency to over-predict coalescence, which may lead to excessive clustering of bubbles and formation of a huge cap within the computational domain. As coalescence has been experimentally identified as a rather scarce phenomenon, and as a statistical steady-state is required to extract averages, we have chosen to prevent coalescence by introducing a small artificial repellent force when bubbles get too close to a wall or are about to coalesce. In other words, the flow considered in this paper can be viewed as a particular state in the development of a fully-established two-phase turbulent flow. In actual fact, coalescence has been prevented by additional ad hoc short-range forces applied on interfaces

Fr ¼ /r rvl

dr dmin with /r ¼ max Ir ;0 dr

ð8Þ

where dmin is the minimal distance towards other bubbles. The intensity of the force Ir ¼ 60N and the range dr ¼ 0:6 mm have been empirically fitted to prevent coalescence. This force is zero when there is no other interface within the range dr . Besides, similar treatment was applied at the walls in order to avoid wall contact (with smaller values of intensity Iw ¼ 40 N and range dw ¼ 0:16 mm). This modification is artificial and leads to a flow that cannot be reproduced experimentally. The behaviour of the flow may be


5


more representative of flows with higher Reynolds numbers, not yet achievable via DNS. In our quest for a better understanding of two-phase flows, a step-by-step approach suggests to keep the most significant physical phenomena (interfacial and turbulent transfers) and to put aside coalescence. It is a common practice in approaching complicated multi-physics phenomena to isolate different elements of the process and study them separately. In this work, we follow this approach to isolate the effect of buoyancy forces on the flow dynamics from other aspects related to liquid film drainage and bubble sliding at a wall. As a consequence, the bubble diameter is controlled. This feature is very useful to compare the DNS results to averaged models because we can focus on the modelling of turbulence and interfacial forces without perturbation from other models. In fact, averaged models for coalescence and break-up are rather complicated and their prediction can vary significantly from one model to another. This simplification enables to trace back the differences with two-fluid models to one single source: the turbulence modelling hypotheses. Besides, the equilibrium between coalescence and break-up is expected to be a rather slow process. As a consequence, the averaging period that would be required to suppress the low frequency variations might be incompatible with a reasonable computational cost. Here, the flow can reach a statistically steady state with a prescribed equivalent bubble diameter and we can analyse the effect of the bubbles on the wall friction and on the averaged flow quantities.

3.3. Validation: single-phase turbulence Several improvements of the computational code were necessary to perform the DNS presented here. Developments were validated mostly on two problems: a single-phase turbulent flow and the terminal velocity and shape of an isolated bubble. The latter is not presented here. Before introducing the bubbles, a fully developed single-phase turbulent channel flow at the Reynolds number Res ¼ 180 was computed. The size and resolution of the computational domain are the same as for the two-phase calculation presented in Table 1. The computed mean velocity and rms fluctuations agrees well with the reference results of Kim et al. (1987) as shown in Fig. 2. Thus, this simulation demonstrates that the computational domain and the mesh resolution are sufficient to compute high-order statistics. Moreover, the correct behaviour of the 3rd order time

differentiation and the 4th order space differentiation is assessed and this validation case also provides a verification of the computation of time- and plane-averaged statistics. 4. The two fluid model In this section, we present the relevant information on the derivation of the averaged equations and on the constitutive equations required to close the two-fluid system. A detailed derivation of the two-fluid model can be found for example in Ishii (1975). The assumptions used in the averaged RANS CFD code NEPTUNE_CFD are presented with a focus on the Reynolds stresses modelling (Section 4.3). In particular, Section 4.5 details the models for interfacial production and dissipation used in NEPTUNE_CFD as their effect on the Reynolds stresses prediction is thoroughly analysed in Section 5.1.2. These models are in fact very influent in twophase flows and they are chosen as the main interest of the upscaling methodology presented in Section 5.1.2. In this paper, we focus on adiabatic bubbly flows and micro-to-macroscale model up-scaling is applied to turbulence. Thus the phase-change phenomena and the energy equation are not considered. Besides, physical properties of each phase are assumed constant. 4.1. Averaging operator The two-fluid model formulation rely on an averaging operator applied to the microscopic equations of mass (1) and momentum (2) presented in Section 3.1. The most classical averaging is ensemble (or statistical) averaging operator (Ishii, 1975), noted with an overbar /. In this work, the following practical definition is used

/ðy; tÞ¼ ^

1 DtLx Lz

Z

tþDt=2

tDt=2

Z 0

Lx

Z

Lz

/ðx; y; z; sÞdxdzds

ð9Þ

0

where Dt is the time averaging period, chosen sufficiently large to make /ðy; tÞ practically independent of time. Averaging the vapour indicator function vv ¼ 1 vl leads to the definition of the void fraction a ¼ av ¼ vv . An other useful definition for the quantities defined per unit mass of phase k (like the velocity which is the momentum per unit mass) is the phase average defined by:

/k k ¼ ^

vk /k vk /k ¼ vk ak

ð10Þ

Fig. 2. Validation of the simulation tool against the DNS of a turbulent channel flow at Res ¼ 180 (Kim et al., 1987).


6


With the previous definition, any quantity can be decomposed into an averaged and a fluctuating part. For example, for the velocity, one gets: uk ¼ ukk þ u0k . For the interfacial terms, the interfacial area concentration (IAC) is defined by ai ¼ di and is used to weight averaged interfacial quantities.

/k i ¼ ^

/k di ai

ð11Þ

4.2. The two fluid model The averaged equations describing the evolution of each phase are obtained from the Navier–Stokes equations written for phase k multiplied by the phase indicator function and then averaging (Delhaye, 1974; Drew and Passman, 1999; Delhaye, 2008; Ishii and Hibiki, 2010; Morel, 2015). Due to the commutability of the averaging operator with the space- and time-derivatives, we obtain the following averaged equations

r ak ukk ¼ 0

ð12Þ

@ ak qk ukk þ r ak qk ukk ukk ¼ rak pkk þ r ak Dtot þ ak qk g þ Mk k @t

ð13Þ

where Dtot ^ sk k qk uk 0uk 0k ; sk ¼ lk ruk þ rT uk is the viscous k ¼ stress tensor, Rk ¼ uk 0uk 0k the turbulent Reynolds stresses and Mk the interfacial momentum transfers (from the interface to phase k) defined by:

Mk ¼ Mi!k ¼ pk rvk sk rvk

ð14Þ

From the microscopic jump conditions (Kataoka, 1986; Delhaye, 1974), we know that

X Mk ¼ Mr ¼ ^ rjnv di

ð15Þ

k

where r is the surface tension and j ¼ 2Hlv ¼ rs nv is the local curvature of the interface (negative for spherical bubbles: j ¼ 2=rb with rb the bubble radius) and Mr is the mixture momentum source due to the surface tension effect. Following the proposal of Ishii and Hibiki (2010), the macroscopic interfacial momentum transfer term Mk is decomposed in the absence of phase-change using the surface mean values (noted /ki ) into

Mk ¼ Mik þ pkki rak þ rak ski k

ð16Þ

where the total generalised drag force Mik (following the notations of Ishii and Hibiki (2010)) is an unknown parameter modelled on the vapour side (Miv ) as a series of forces. In the context of the twofluid one-pressure model of dispersed bubbly flows, we assume pkki ¼ pvv ¼ pll and the effect of the interfacial shear ski k and the mixture momentum source Mr are often neglected in a first approximation. In the end, the main variables av ; ull ; uvv and p describing the evolution of the averaged thermodynamic system are governed by Partial Differential Equations for the vapour, for the total mass and for the liquid and vapour momentum conservation

@ av þ r av uvv ¼ 0 @t

@ al ql þ av qv ð17Þ þ r al ql ull þ av qv uvv ¼ 0 @t l

@ al ql ul þ r al ql ull ull ¼ al rp þ r al Dtot þ al ql g Miv l @t v

@ av qv uv þ r av qv uvv uvv ¼ av rp þ r av Dtot v þ av qv g þ Miv @t supplemented by algebraic constitutive relations to define secondary variables: al ¼ 1 av ; u0l u0l l ; u0v u0v v and Miv . For these vari-

ables, dedicated constitutive equations are required in order to fully close the system. The liquid Reynolds stresses Rij ¼ u0l u0l l are modelled by a second order turbulence model (Rij e) whereas the vapour Reynolds stresses are neglected: u0v u0v v 0. In single-phase flow, the closure issue is limited to the modelling of the turbulent Reynolds stresses. In two-phase flow, the problem is made more complex by the interactions between phases that produce momentum transfer from one phase to the other, or energy accumulation and redistribution on interfaces by means of the surface tension. Regarding the forces modelling the momentum transfer Miv , the model of Ishii and Zuber (1979) is used to model the drag force. The virtual mass and Basset forces are accounted for by means of Zuber’s model (Zuber, 1964). The lift and wall lubrication forces are presented in Tomiyama (1998) and Tomiyama et al. (2002). Lastly, the model for turbulent dispersion force is given in Laviéville et al. (2015). Then, the set of Eq. (17) can be solved in averaged codes to predict the flow evolution.

4.3. Governing equations for the liquid Reynolds stresses In two-phase flow, the modelling of the liquid Reynolds stresses classically rely on single-phase models, later extended to account for some effect of the void fraction. One of the most complete form of closure for the Reynolds stresses is the Reynolds Stress Model (RSM) as it retains the anisotropy of turbulence. The exact transport equations for the Reynolds stresses Rij are derived from the local momentum equations, first writing an equation for the velocity fluctuation u0i , then multiplying it by u0j and finally averaging (Lance, 1979; Lance et al., 1984; Lance, 1986)

@ al Rij @ al ulb Rij ¼ al P ij al ij þ al Uij þ @t @xb

a

@ @ al T ijb ml al Rij þ l p0 u0i l dbj þ p0 u0b l dib @xb @xb ql

l 1 0 0 p uj ni þ p0 u0i nj di

ql

2

3 l l @u0i u0j @ i i 0 0 þ ml 4 u u nj d þ nb d 5 @xb i j @xb

ð18Þ

where dij is the Kronecker delta equal to one if i ¼ j or 0 otherwise. The production P ij , the triple velocity correlation tensor T ijb , the pressure strain correlation tensor Uij and the dissipation tensor ij are defined by:

Pij ¼ Rib

@ulj @xb

þ Rjb l

@uli @xb

T ijb ¼ u0i u0j u0b 0 l p0 @u0i @uj Uij ¼ þ ql @xj @xi

ij ¼ 2

ll @u0i @u0j l ql @xb @xb

ð19Þ ð20Þ ð21Þ ð22Þ

The term Pij is the production of the correlation Rij by the mean velocity gradient and needs no additional modelling. As the triple velocity correlation tensor T ijb appears in a divergence, it is only responsible for transport of the correlation Rij but by the fluctuating velocity field. It needs to be modelled. The pressure strain correlation tensor Uij also requires some modelling efforts. For high Reynolds number flows, as a consequence of the local isotropy, the dissipation tensor ij is often assumed isotropic in singlephase flow


7


2 3

ij ¼ dij

ð23Þ

with the following definition for the scalar dissipation rate :

al ¼ ml vl

@u0i @u0i @xj @xj

ð24Þ

One last equation then needs to be derived and closed to describe the evolution of the turbulent dissipation rate . 4.4. Modelling of the liquid Reynolds stresses The RSM model presented by Speziale et al. (1991) (hereafter noted SSG) is a very common reference to model the pressure redistribution term. Besides, the effect of the void fraction is considered in our model by specific interfacial production terms for the Reynolds stresses and for the turbulent dissipation. Their model is briefly presented in Section 4.5. The model can be used along with a wall-law for the liquid velocity that accounts for the roughness generated by the bubbles at the wall (Mimouni et al., 2010). At low Reynolds number, the first discretization point easily falls into the buffer region or the viscous sublayer, hence reducing the effect of the law-of-the-wall. As a consequence, the closure of the RSM equations are used in the neighbourhood of the wall where the turbulence anisotropy prevails and the wall-echo effect is important. Even if the wall-law can be used in the buffer layer or in the viscous sublayer, the SSG model is not suitable to predict the anisotropy of turbulence in the near-wall region. Bradshaw et al. (1987) assessed the validity of the local approximation for the rapid pressure term (used in the SSG model) using the DNS of Kim et al. (1987). They showed that this hypothesis is valid only for yþ P 40. As a result, models based on this assumption cannot be integrated down to the wall without modifications. The closure of the redistribution tensor Uij must be modified to model the wallecho effect (Gibson and Launder, 1978; Gerolymos et al., 2012). In order to avoid the ad hoc damping functions, which are usually calibrated on experimental or DNS data with little theoretical background, we have preferred to implement the Elliptic Blending Reynolds Stress Model (EBRSM) proposed by Manceau and Hanjalic´ (2002) and later modified by Manceau (2005). This model affects the redistribution term in order to satisfy the near-wall balance

/I ij ij ¼ ml

@ 2 Rij @x2b

ð25Þ

where /I ij models the pressure–velocity gradient correlation Uij . This constraint is fulfilled by a blending of the standard SSG model w (/hij hij , valid far from the wall) with near-wall models (/w ij ij , satisfying the asymptotic behaviour in the near-wall region). The elliptic blending factor aeb varying from 0 at the wall to 1 far from the wall used as follows

/I ij ij

w ¼ 1 a3eb /w þ a3eb /hij hij ij ij

ð26Þ

preserves the non-local character of the blocking effect and is obtained from the resolution of the elliptic relaxation equation

aeb L2 r2 aeb ¼ 1 with ajw ¼ 0 where L ¼ C L max

3=2

k

3=4

; C g m1=4

ð27Þ

is Durbin’s length scale.

Contrary to the simpler SSG model, the EBRSM is able to predict the anisotropy of the Reynolds stresses and the non-local effect of the pressure fluctuations in the neighbourhood of the wall including the buffer layer and viscous sublayer.

4.5. Interfacial production The interfaces are responsible for additional correlations appearing on the last line of Eq. (18). Experimental observations tends to indicate that energy exchange between the different Reynolds stresses towards isotropy is reinforced by the presence of the bubbles. Besides, the liquid turbulence energy produced in the wakes of bubbles is immediately dissipated (Morel, 2015). In order to consider the modification of turbulence by the bubbles, interfacial production terms for the Reynolds stresses Piij and for the turbulent dissipation Pi are considered in the Rij and respectively. They are modelled as

Piij ¼ 2=3al av bij m Pi ¼ tr Piij =s

equations

ð28Þ ð29Þ

where bij ¼ 2=3dij þ Vri Vrj =kVr k2 is the anisotropic distribution ten-

sor, Vr ¼ uvv ull is the relative velocity and m is the limiter forcing the production by the relative velocity mv r to remain smaller than the local turbulent dissipation rate m

m ¼ mmin ¼ minðmv r ; m Þ with mv r ¼ C D kVr k2 m ¼ ql =av

and ð30Þ

The timescale s in Eq. (29) is defined as the maximum between the turbulence integral timescale st and the bubble/turbulence interaction timescale sb 1=3 s ¼ maxðst =C 2 ; sb Þ with st ¼ k= and sb ¼ D2=3 ð31Þ b

where k ¼ 1=2Rii is the turbulent kinetic energy and C 2 ¼ 1:88 according to Manceau and Hanjalic´ (2002). In the next section of this article, the influence of the closure hypotheses (28) and (29) is assessed. 5. Two fluid simulations In the scope of two-phase flow modelling, detailed experimental data are rather scarce because of technical limitations to access local quantities. Nonetheless, local measurements are essential to develop constitutive relations closing the turbulent and interfacial transfers, thereby closing the two-fluid model. An other way to contribute to the improvement of the two-fluid model is to use DNS as a reference to compute the averaged quantities required by the two-fluid model from the local fields. This methodology (called ‘‘up-scaling”) rely on local simulations (assimilated to ‘‘numerical experiments”) to provide information to larger scale models. 5.1. Averaged simulations and mesh description Under the ergodic hypothesis, the problem considered in Section 3 can be averaged, after the transitional regime, over time and homogeneous directions to lead to a one-dimensional steady-state at the averaged scale described by two-fluid models. Therefore, the averaged description of the problem is assessed on a 1D grid. Symmetry boundary conditions are used at the centre of the channel and in the spanwise direction whereas periodic conditions are imposed on the faces normal to the streamwise direction. In the streamwise direction, the flow is generated by a source term ak Sk ¼ qm g x sw0 =h compensating for the wallfriction and for the hydrostatic pressure in such a way that the numerical pressure gradient is nil in the direction of the flow. The case is studied in NEPTUNE_CFD (Guelfi et al., 2007) with a row of hexahedral cells as represented on Fig. 3.


8


reference results from Kim et al. (1987) for the single-phase flow and to our DNS results for the two-phase flow. The profile of the mean velocity non-dimensionalized by the wall-shear velocity us is shown in Fig. 4.

Fig. 3. Description of the numerical domain used for NEPTUNE_CFD calculations.

Fig. 4. Averaged single-phase (1/) and two-phase (2/) flows: comparison of liquid velocities with EBRSM or SSG model to DNS references (either present calculation or Kim et al., 1987).

Single- and dispersed-two-phase flow predictions are compared with the standard SSG or the EBRSM turbulence modelling for low Reynolds conditions. Two cases were computed: (i) the DNS condition with 10% void fraction constituted by monodispersed bubbles with a diameter of 1 mm and (ii) the single-phase flow conditions computed in the DNS of Kim et al. (1987). Both cases involve a low Reynolds number (Res ¼ 180) and the predictions of the two turbulence models (SSG and EBRSM) are compared to evaluate the current state of the code models. Mesh refinement has been performed along the wall-normal direction using a uniform resolution. The converged results obtained for a resolution of N y ¼ 160 cells are compared to the

5.1.1. Single-phase flow SSG modelling For the SSG modelling of the single-phase flow configuration, the velocity of the first element near the wall is accurately computed and matches the law of the wall even when the cell-centre is within the laminar sublayer, yþ < 5 where uþ ¼ yþ . Besides, the slope of the velocity profile in the logarithmic region satisfactorily matches the one of the reference solution but the velocity is strongly under-estimated because the turbulence model is unable to predict the velocity evolution in the buffer layer. As a result, the maximal and bulk velocities are underestimated by 40%. The important error on the flow-rate is also reported in Table 3 and compared to reference values. Consequently, the analysis of low-Reynolds flow cannot be achieved with the SSG model. Therefore, some key features of the validation of the low-Reynolds EBRSM model are presented hereafter. EBRSM modelling The EBRSM model has been implemented in NEPTUNE_CFD (Bois, 2015). Converged results are shown on Figs. 4–6. Very good agreements with the reference results of Kim et al. (1987) are observed using a regular mesh with 160 cells. Major improvements are clear on any quantity. The velocity is in excellent agreement with the DNS (see Fig. 4). Very good agreement is also obtained for the Reynolds shear-stress Rxy , the axial Rxx and the wallnormal stresses Ryy (Fig. 5a). Only the spanwise stress Rzz is slightly over-estimated. The EBRSM prevents the divergence of the Reynolds stresses and turbulent dissipation in the viscous sublayer as yþ goes to 0. It is particularly important to have correct predictions to ensure that the turbulent dissipation remains finite which is the case with the EBRSM, whereas it was divergent with the standard SSG model (not shown). Another important feature of the EBRSM is the correct limiting behaviour of the crosscorrelation Rxy : limyþ !0 Rxy ¼ 0. This theoretical limit is not recovered by the standard SSG model.

5.1.2. Two-phase flow SSG modelling For the two-phase flow, Fig. 4 shows that the velocity of the first element near the wall matches the law of the wall uþ ¼ yþ when mesh convergence is achieved. The velocity profile exhibits a negative gradient @U=@y in the region yþ 2 ½2; 10 and predicts a flow reversal around yþ ¼ 10. This local minimum and the negative axial velocity are in total disagreement with the DNS solution and no physical principle or experiment has been found to support it. Therefore, the SSG model is not satisfactory as it is unable to evaluate correctly the two-phase flow. The flow rate prediction of the two-phase flow simulation is rather similar to the single-phase flow case (see Table 3).

Table 3 Mean flow comparison of NEPTUNE_CFD simulations to reference results provided by Kim et al. (1987) for single-phase flow and the present DNS in two-phase flow. Flow

Model

Rec ¼ Umclh

m Rem ¼ 2hU ml

U m =us

U c =us

U c =U m

Single phase

Ref. SSG EBRSM

3300 2272 3352

5600 3712 5751

15.63 10.31 15.98

18.20 12.62 18.62

1.16 1.22 1.17

Two phase

Ref. SSG EBRSM

1518 2396 3824

3391 4020 6472

12.72 11.17 17.98

14.54 13.31 21.25

1.20 1.19 1.18



9

Fig. 5. Predictions of Reynolds stresses with EBRSM or SSG model for single-phase or two-phase flows compared to DNS references. Symbols are for DNS references, lines for the SSG model predictions and dotted-lines for the EBRSM. Each colour is associated to a component: black, red, green and blue are for Rxx ; Ryy ; Rzz and Rxy respectively.

Fig. 6. Predictions of void fraction and averaged velocities with EBRSM or SSG model for two-phase flow compared to DNS reference.

Compared to the DNS results, it is underestimated by around 15% in that configuration. EBRSM modelling with standard interfacial production A preliminary evaluation of the performance of the EBRSM in the two-phase flow conditions of the DNS has been conducted. The standard formulation for the interfacial productions Piij and P i presented in Section 4.5 is used. The liquid velocity (Fig. 4) and the Reynolds stresses (Fig. 5b) are compared to the reference results given by the DNS and to the standard SSG model. Fig. 6 shows the void fraction and velocities profiles. The turbulence model has a very limited effect on the void fraction prediction. The intensity of the wall peak is captured rather accurately but the shape and the location of this maximum could be improved working in particular on the formulation of the lubrication wall force. Indeed, in the near-wall region, the wall force model of Tomiyama leads to zero void fraction up to yþ 15 while in the DNS, the void fraction starts increasing when yþ > 2 even with the wall forces used. Further away from the wall peak, the void

fraction diminishes very smoothly in NEPTUNE_CFD simulations, reaching a value very close to the mean void fraction at the channel centre line. The DNS profile tends more rapidly to this averaged value and secondary oscillations are clearly visible. They are related to the bubble size and correspond to bubble accumulation at a given distance away from the wall. This fine description is smoothed by NEPTUNE_CFD models and only the global decreasing trend is captured. Moreover, neither model is able to predict accurately the liquid velocity profile in the bubbly flow. The EBRSM behaves better in the viscous sublayer and buffer region whereas the standard SSG model shows a better agreement with the DNS data in the core of the flow. The predictions of the diagonal components of the Reynolds stresses are satisfactory, especially with the standard SSG model but unfortunately, the shear stress is poorly predicted by both models in the buffer region where the peak occurs. To conclude, the major improvements brought by the EBRSM in the case of single-phase flow have not been as significant on the


10


Fig. 7. Parametric studies on the Reynolds stresses predictions with the EBRSM for two-phase flow compared to DNS reference. sstd is the value given by Eq. (31). mmin and mv r are defined by Eq. (30).

two-phase flow configuration. Thus, in the current status of the analysis, the modelling choice is not obvious and some modifications have to be proposed. This flow example has been designed to focus on turbulence coupling in two-phase flows. In fact, our analysis indicates that the interfacial transfer modelling plays a predominant role over turbulence modelling in the flow prediction. The model for the generalised drag force is not accurate enough to perform finer analysis on the RSM closure. EBRSM modelling parametric study on interfacial production To perform a finer analysis on the turbulence closure hypothesis, it is thus necessary to split the problem in two parts and to run simulations where the velocities and the void fraction are prescribed from DNS. Thus, the effect of the interfacial momentum exchange term is eliminated and turbulence models are assessed separately. As the SSG model is unable to reproduce the reference results in single-phase flow, our work is focused on the EBRSM. Only interfacial production terms are modified because the good predictions of the model in single-phase flow should be preserved. The results are presented on Fig. 7. With the standard model presented in Section 4, the strong overestimation of the liquid velocity leads to too high values of the Reynolds stresses. When the coupling between the mass and momentum equations and the RSM equations is eliminated (prescribing void fraction and velocities from DNS), the interfacial production in the standard model is strongly restrained by the numerical conditioning (30), leading to very low values of the Reynolds stresses. When interfa-

cial production is unbridled (using mv r instead of mmin in Eq. (30)), high values of the diagonal Reynolds stresses are predicted whereas satisfactory estimations are obtained for the crosscorrelation Rxy . The level of turbulence is then directed by the dissipation timescale. The definition (31) leads to the numerical value s 0:02 s. Halving this timescale leads to a much better prediction of the diagonal Reynolds stresses at the expense of the cross-correlations. Future work is necessary to improve the prediction of all Reynolds stresses. It could be achieved for instance by an anisotropic definition of the turbulence dissipation rate tensor.

6. Conclusion A Direct Numerical Simulation of a turbulent bubbly flow in a vertical channel has been performed and thoroughly analysed in view of improving turbulence models for two-fluid RANS CFD. Physical properties of saturated steam and water at high pressure (P ¼ 155 bars) were used. The flow is strongly affected by buoyancy forces that lead to a very strong relative velocity. The liquid velocity is even flatter than in single-phase flows. Neither wall nor core peaking of the void fraction appears. The velocity fluctuations strongly enhanced by the bubbles are no longer limited to the near-wall layer. In this context, it is not clear that the application of single-phase turbulence modelling to such a flow is predictive, even if the model has been adapted to try and take into account the effect of the void



fraction. Time- and space-averaging in the periodic directions has been performed on the local fields predicted by DNS. The outcome has been used in several ways in order to assess the quality of closure hypotheses used at the averaged scale in the two-fluid RANS CFD model implemented in NEPTUNE_CFD. For the adiabatic two-phase flow with mono-dispersed bubbles considered here, relevant models account for turbulence and interfacial momentum transfer formulated under the form of a generalised drag force. The current level of modelling for the interfacial momentum transfer has proven insufficient to accurately predict the void fraction and velocity profiles. It has a predominant impact on flow predictions over turbulence modelling. Nonetheless, in a step-by-step approach, the transport equations of the Reynolds stresses have been uncoupled from the Navier–Stokes equations and investigated separately. At low-Reynolds, a turbulence closure valid in the buffer layer where the pressure wall-echo term is important is necessary. A low-Reynolds turbulence model (EBRSM) has been implemented in NEPTUNE_CFD and validated against the results of single-phase flow DNS. Striking improvement is observed compared to the standard SSG model. Interfacial productions of turbulence and dissipation have been incorporated in the model to account for the modification of turbulence by the bubbles and the relative velocity. In first approximation, these productions control the solution. Ways of improvements of these two models have been investigated but no set of parameters was found capable of predicting all the components of the Reynolds stresses yet. Our work suggests to revise the formulation of the turbulent relaxation timescale in the transport equation of the turbulent dissipation. In fact, we showed that different timescales are required to predict the correct level for diagonal and off-diagonal Reynolds stresses. This suggests to look for a modification in the formulations of the models that better accounts for the anisotropy of the pressure redistribution term and of the turbulence dissipation rate. Further analysis of these DNS results towards modelling of the interfacial transfer is under way (Bois et al., 2016). Besides, this DNS could also be analysed in view of developing the Interfaces and Subgrid Scales (ISS) modelling (Toutant et al., 2009; Toutant et al., 2009; Bois et al., 2010; Bois, 2011) in order to decrease the computational cost and facilitate future computations for higher Reynolds numbers or for parametric studies. Aknowledgement The author would like to thank the NURESAFE project for funding this research. The author would also like to convey his sincere thanks to GENCI and the TGCC for providing the necessary computational resources to perform DNS calculations. This work was granted access to the HPC resources of TGCC under the allocation 20XX-t20142b7239 made by GENCI. References Aulery, F., Toutant, A., Bataille, F., Zhou, Y., 2015. Energy transfer process of anisothermal wall-bounded flows. Phys. Lett. A 379 (24–25), 1520–1526. Bois, G., 2011. Transferts de masse et d’énergie aux interfaces liquide/ vapeur avec changement de phase: proposition de modélisation aux grandes échelles des interfaces (Ph.D. thesis), Université de Grenoble. Bois, G., 2015. Report on upscaling and comparison of momentum transfer models. NURESAFE D23, 41. Bois, G., Jamet, D., Lebaigue, O., May-June 2010. Towards Large Eddy Simulation of two-phase flow with phase-change: Direct Numerical Simulation of a pseudoturbulent two-phase condensing flow. In: Proceedings of the 7th International Conference on Multiphase Flow (ICMF ’10). Tampa, Florida, USA. Bois, G., Mathieu, B., Fauchet, G., Toutant, A., May 2016. DNS of a turbulent steam/ water bubbly flow in a vertical channel. In: Proceedings of the 9th International Conference on Multiphase Flow (ICMF-2016). Firenze, Italy. Bradshaw, P., Mansour, N.N., Piomelli, U., 1987. On local approximations of the pressure-strain term in turbulence models. In: Proceedings Summer Program. Center for Turbulence Research, NASA Ames/Stanford Univ, pp. 159–164.

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