Density stratification breakup by a vertical jet: Experimental and numerical investigation on the effect of dynamic change of turbulent schmidt number

The hydrogen behavior in a nuclear containment vessel is one of the significant issues raised when discussing the potential of hydrogen combustion during a severe accident. Computational Fluid Dynamics (CFD) is a powerful tool for better understanding the turbulence transport behavior of a gas mixture, including hydrogen.

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Density stratification breakup by a vertical jet: Experimental and numerical

investigation on the effect of dynamic change of turbulent schmidt number

Satoshi Abea,⁎, Etienne Studerb, Masahiro Ishigakia, Yasuteru Sibamotoa, Taisuke Yonomotoa

a Thermohydraulic Safety Research Group, Nuclear Safety Research Center, Japan Atomic Energy Agency, 2-4, Shirakata-Shirane, Tokai, Ibaraki 319-1195, Japan

b DEN-STMF, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

A R T I C L E I N F O

Keywords:

Density stratification

Nuclear containment vessel

RANS

Turbulent Schmidt number

A B S T R A C T The hydrogen behavior in a nuclear containment vessel is one of the significant issues raised when discussing the potential of hydrogen combustion during a severe accident Computational Fluid Dynamics (CFD) is a powerful tool for better understanding the turbulence transport behavior of a gas mixture, including hydrogen Reynolds- averaged Navier–Stokes (RANS) is a practical-use approach for simulating the averaged gaseous behavior in a large and complicated geometry, such as a nuclear containment vessel; however, some improvements are

re-quired In this paper, we focused on the turbulent Schmidt number Sc t for improving the RANS accuracy Some

previous studies on ocean engineering mentioned that the Sc t value gradually increases with the increasing

stratification strength We implemented the dynamic modeling for Sc t based on the previous studies into the OpenFOAM ver 2.3.1 package The experimental data obtained by using a small scale test apparatus at Japan Atomic Energy Agency (JAEA) was used to validate the RANS methodology In the experiment, we measured the velocity field around the interaction region between vertical jet and stratification by using the Particle Image Velocimetry (PIV) system and time transient of gas concentration by using the Quadrupole Mass Spectrometer (QMS) system Moreover, Large-Eddy Simulation (LES) was performed to phenomenologically discuss the in-teraction behavior The comparison study indicated that the turbulence production ratio by shear stress and

buoyancy force predicted by the RANS with the dynamic modeling for Sc t was a better agreement with the LES result, and the gradual decay of the turbulence fluctuation in the stratification was predicted accurately The time transient of the helium molar fraction in the case with the dynamic modeling was very closed to the VIMES experimental data The improvement on the RANS accuracy was produced by the accurate prediction of the turbulent mixing region, which was explained with the turbulent helium mass flux in the interaction region Moreover, the parametric study on the jet velocity indicates the good performance of the RANS with the dynamic

modeling for Sc t on the slower erosive process This study concludes that the dynamic modeling for Sc t is a useful and practical approach to improve the prediction accuracy

1 Introduction

As emphasized in the Fukushima–Daiichi accident, the hydrogen

behavior raises concern for the safety of a light water reactor (LWR) (

Breitung and Royl, 2000; Lopez-Alonso et al., 2017, OECD/NEA, 1999)

During a severe accident in an LWR, a large amount of hydrogen gas

can be produced by the metal/steam reaction and released in a nuclear

containment vessel To understand the mechanism underlying these

hydrogen transport phenomena, nuclear research groups have

per-formed experimental and numerical studies on the stratification

breakup behavior using several types of jets

Computational fluid dynamics (CFD) analysis is a powerful tool for

better understanding the hydrogen transport behavior in a nuclear

containment vessel Thus, many CFD benchmark tests have been con-ducted under the auspices of Organisation for Economic Co-operation and Development/Nuclear Energy Agency (e.g., international standard problem No 47 (ISP-47) (Allelein et al., 2007; Studer et al., 2007), the SETH project (Auban et al., 2007), the SETH-2 project (OECD/NEA Committee on the Safety of Nuclear Installations, 2012), and the third international benchmark exercise (IBE-3) (Andreani et al., 2016)) The experimental condition for the IBE-3 conducted in the PANDA facility (Paladino and Dreier, 2012) was designed to investigate the stratifica-tion erosion by a vertical jet from below These benchmark tests in-dicated that the turbulence model is an important factor in the accurate prediction of hydrogen transport and distribution (Kelm et al., 2019) Considering the computational cost and time, the Reynolds-

https://doi.org/10.1016/j.nucengdes.2020.110785

Received 10 February 2020; Received in revised form 15 May 2020; Accepted 28 July 2020

⁎Corresponding author

E-mail address: abe.satoshi@jaea.go.jp (S Abe)

Available online 02 September 2020

0029-5493/ © 2020 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

T

averaged Navier–Stokes (RANS) approach is a practical tool for

simu-lating the averaged gaseous behavior in a large and complicated

geo-metry, such as a real nuclear containment vessel In the OECD/NEA

HYMERES (Hydrogen Mitigation Experiments for Reactor Safety)

pro-ject, the k–ε model (Launder and Spalding, 1974) was used as a

“common model” (Studer et al., 2018; Andreani et al., 2019) because of

its low computational cost and good numerical stability The buoyancy

effect on the turbulence property must be accurately estimated to

im-prove the accuracy of RANS modeling on the stratification behavior

(Kelm et al., 2019; Abe et al., 2015) In a collaboration research activity

of the Commissariat à l'énergie atomique et aux énergies alternatives

(CEA) and the Japan Atomic Energy Agency (JAEA), we performed a

CFD simulation on the HM1-1 benchmark in the OECD/NEA HYMERES

project (Studer et al., 2018) In this benchmark test, we focused on the

change of the turbulent Schmidt number Sc t and Prandtl number Pr t in

the stratification, which are usually set to constant values of less than

unity (Ishay et al., 2015; Tominaga and Stathopoulos, 2007) and

di-rectly affect the turbulence production and turbulence transport

beha-vior However, some recent studies on ocean engineering

(Venayagamoorthy and Stretch, 2010; Elliott and Venayagamoorthy,

2011; Strang and Fernando, 2001) mentioned that these numbers

dy-namically change with the increase of the stratification strength We

implemented the dynamic modeling for the turbulent Schmidt number

Sc t and Prandtl number Pr t based on the formulation developed by

Venayagamoorthy and Stretch (2010) Consequently, the CFD result

was in a good agreement with the MISTRA experimental data,

in-dicating that the accuracy was improved by changing Sc t and Pr t (Abe

et al., 2018a,b) Additionally, our study mentioned that a further

vali-dation with detailed experimental data on a simpler condition is

re-quired

A small-scale experiment is one useful approach for obtaining the

detailed experimental data for the CFD validation Deri et al (2010)

measured the velocity field around the interaction region between a

vertical jet and stratification in a small-sized rectangular vessel We at

the JAEA also constructed a small experimental apparatus, called the

VIsualization and MEasurement system on Stratification behavior

(VIMES), to observe the gaseous mixture behavior in a rectangular

vessel (Abe et al., 2016) The objectives of the VIMES experiment is to

visualize the flow field with the Particle Image Velocimetry (PIV)

system and measure the time transient of the helium concentration at

several locations Several types of obstacles were installed in the test

vessel, and the interaction behavior between the complicated flow and

stratification was investigated (Abe et al., 2018a,b) In this study, the

data from the VIMES test are used for the dynamic modeling for Sc t

Combined with experimental research, the Large -Eddy Simulation (LES) can provide more insights into the turbulence phenomena; however, it is not realistic to apply it for simulating the gaseous be-havior in a real containment vessel because of the necessity of high computational cost and time Röhrig et al (2016) performed the LES and RANS on a light gas stratification breakup by a vertical jet in the small-scale vessel conducted by Deri et al (2010) at the CEA This re-search concluded that the LES yields a decent prediction of the char-acteristic erosion process Moreover, they mentioned that the RANS approaches manage to capture the overall behavior though with a no-table lack in accuracy, indicating the improvement of the RANS accu-racy is required Sarikurt and Hassan (2017) used the LES methodology

on the IBE-3 in the PANDA facility and investigated the flow structures for the interaction of a buoyant jet and a stratified layer The research summarized that understanding the interaction mechanism will help quantify the turbulent mass transfer of the gas component In this study,

we performed the LES to obtain detailed turbulence properties in the interaction region, such as turbulence fluctuation and turbulent mass flux

This paper phenomenologically discusses the interaction behavior between a vertical jet and stratification The capability of the dynamic

modeling for Sc t in RANS is shown based on a phenomenological un-derstanding The VIMES experimental data are used to validate the LES and RANS The remainder of the paper is organized as follows: Section

2 describes the VIMES apparatus with the initial and boundary condi-tions and the brief experimental results; Section 3 presents the nu-merical and boundary conditions, including the turbulence model, mesh, and discretization schemes, and explains the validity confirmed

by various perspectives (i.e., mean and turbulence profiles, velocity spectra, and mesh convergency); Section 4 shows a comparison of the simulation result with the experimental results, and turbulence pro-duction phenomena obtained by the LES and RANS; Section 5 presents the turbulence mixing behavior in the interaction region between the vertical jet and stratification, and a parametric study to evaluate the

capability of dynamic modeling for Sc t; and Section 6 summarizes the main conclusions

2 VIMES apparatus

The VIMES apparatus had a rectangular acrylic vessel with 1.5 m width, 1.5 m length, and 1.8 m height (Fig 1) Two horizontal nozzles with 0.03 m diameter were inserted for injecting the binary gas of air and helium (as a mimic gas of hydrogen) controlled with two mass flow controllers An upward nozzle with 0.03 m diameter was equipped to

Fig 1 Schematic of the VIMES apparatus (a) schematic of gas line and test vessel and (b) photograph of test vessel

inject a vertical jet from the bottom of the test vessel The insert length

Hinj was 0.1 m The flow field was visualized with a two-dimensional

Particle Image Velocimetry (PIV) This system consisted of 135 mJ

Pulsed Nd:Yag laser and a black and white Andor NEO 5.5 camera with

a resolution of 2560 × 2160 px and Nikon 50 mm f/1.2 s The PIV

system measured with an error of less than 4% on the total momentum

flux of a jet at any downstream location The size of the field-of-view

(FOV) achieved approximately 500 mm height and 600 mm wide The

FOV was set to z = 1.0 to 1.5 m (z/D = 33.3 to 50) to observe the

interaction behavior between the jet and stratification The acquisition

rate of the PIV system was set to 8 Hz The gas concentration was

measured using the Quadrupole Mass Spectrometer (QMS) system with

a multiport rotating valve for multipoint measurement The capillary

tubes with 1.0 mm inner diameter were connected to the rotating valve

The pipe ends were placed at the near corner of the test vessel (Fig 1)

The measurement system was validated based on multiple experiments,

as mentioned in detail below

2.1 Initial and boundary conditions

All experiments in this paper were performed under the condition of

iso-thermal Gas temperature in the initial and inlet conditions was

approximately 288.15 (± 5.0) deg-K The binary gas of air and helium

was injected to build up the initial density stratification, as shown in

Fig 2(a) The injection flow rate was 105( 6.0)± L/min The molar

fractions of helium and air were 70% and 30%, respectively The

in-jection duration was 420 s Consequently, the density stratification was

formed above 1.0 m The maximum value of the helium molar fraction

reached approximately 60% at the top of the test vessel A horizontal

bar attached to each data point in Fig 2(a) indicates the standard

de-viation taken from nine experiments conducted with the same initial

conditions, showing that the helium molar fraction was measured with

an error of less than 3% Fig 2(b) compares the time history of the

integrated injection volumetric flow rate derived from the mass flow

rate and the air–helium mixture volumes estimated from the QMS

measurements A good agreement indicates a one-dimensional vertical

distribution of helium gas

At 120 s from the end of the horizontal injection for the

stratifica-tion buildup, the vertical air jet was initiated with the upward nozzle

(Fig 1) to produce the stratification breakup The start time of the jet

injection was defined as Time = 0 s in this paper Table 1 shows the

experimental case The jet velocity was 5.0 ±( 0.15)m/s in the base case

(Case 1) and 2.5 and 3.8 m/s in the parametric cases (cases 2 and 3,

respectively) We performed five tests for the base case and twice for

each parametric case to assess the reproducibility of the VIMES ex-periments The error bars shown in the figures below are the standard deviations from the independent measurements

Studer et al defined the interaction Froude number Fr i to express the interaction behavior between the jet and stratification (Studer et al.,

2012)

=

NL

where the W and L are the velocity and the diameter of the jet in the

impingement region, respectively (Rodi, 1982), and the N is the

char-acteristic pulsation of the stratification These values are defined as follows:

=

6.2 inj

=

=

+

H

2

s

s s

0

where W injin Eq (2) is the velocity magnitude at the nozzle exit, and s and H sare the density and the height of the initial stratified layer,

re-spectively In the VIMES experimental condition, the value of H swas 0.65 m, where the nominal bottom of the initial stratification was as-sumed to z = 1.0 m as shown in Fig 2 Table 1 shows the value of Fr iin each case

2.2 Main experimental result in case 1

Fig 3 shows the visualized flow field with the PIV system in Case 1

at 46 s The color contour shows the velocity magnitude based on radial and vertical components u r2+ w2 This figure indicates the upward jet impingement on the stratification and the rebounding flow surrounding this The occurrence of a strong turbulence mixing was estimated from Fig 4, showing the time transients of the helium gas molar fraction at heights of 0.1, 1.3 1.5, and 1.7 m from the bottom of the test vessel The vertical jet achieved approximately z = 1.3 m (Fig 3); hence, the sharp decay of the helium fraction occurred in the lower region of the initial stratification (line for z = 1.3 m, Fig 4) In the upper region, the slow erosive process was kept before the jet achievement The decrease rate then became faster induced by the strong turbulence mixing Fig 5 shows the vertical distribution of the helium molar fraction The bottom

of the stratification was pushed up, and the volume of the stratification

Fig 2 (a) Vertical distribution of the helium molar fraction at 0 s and (b) time history of the integrated flow of the helium component rate during the stratification

buildup with air–helium gas mixture injection The vertical jet was started at time = 0 s

region gradually decreased This behavior also indicated that the strong

turbulence mixing appeared at the interaction region between the jet

and stratification, and the height of the jet achievement gradually rose

3 CFD simulation

The CFD simulation was performed with the rhoReactingFoam in

OpenFOAM ver 2.3.1 package, an open source code developed by the

OpenFOAM® Foundation The governing equation system in this solver

consists of the continuity, momentum, and transport equations for mass

fraction and enthalpy The detailed description of the momentum and

mass transfer equations is shown below

3.1 LES

The equation governing momentum transport for compressible flow

in the LES is

p

u x

u

i j j i ij

(5)

where u i , , p, and µ are the velocity component in the ith direction,

fluid density, pressure, and molecular viscosity, respectively µ is

cal-culated with the Sutherland equation, consequently corresponding to

approximately 1.8e−05 Pa∙s under the condition of 288.15 deg-K in the

ambient pressure The fourth term at the right-hand side is the

buoy-ancy term g i is the gravity accelation The overline denotes a three-

dimensional space filter operation with a filter width Δ derived with the

cube root of the computational cell volume as follows:

where x, y,and zrepresent the cell size in the respective coordinate direction The Favre density filtering was employed to reduce the complexity of the compressible equation for the LES This operation is expressed with a tilde The subgrid-scale (SGS) tensor ij must be modeled to close the equation system The Boussinesq approximation, assuming a linear correlation between the SGS tensor and the filtered

Table 1

VIMES experimental and simulation cases

Initial Stratification Vertical jet

D = 0.03 m,

H inj = 0.1 m,

ρ 0 = 1.17 kg/m 3

H s =0.65 m

ρ s =0.56 kg/m 3

Fig 3 Instantaneous flow field in the interaction region of the jet and stratification obtained with the PIV measurement in Case 1 The color contour shows the

velocity magnitude based on rdial and vertical components u r2+ w2(m/s)

Fig 4 Time transients of the helium molar fraction (%) in the VIMES

experi-ment at z = 0.1 m, 1.3 m, 1.5 m, and 1.7 m in Case 1 The error bars are the standard deviation from five independent experiments

strain rate tensor = s~ ij 12(u x + u x)

i j j

i , is utilized as follows:

3

In the Smagorinsky model [Smagorinsky, 1963], the SGS viscosity

µ SGSis modeled as

=

In this study, the model constant C s was set to 0.1 based on the

literature (Wang et al., 2006; ANSYS, 2009) The Favre-filtered

trans-port equation of mass fraction is expressed as

Y x

µ Sc

Y x

k i SGS SGS k

where Y k means the mass fraction of kth gas, helium, and air in this

study Ddenotes the diffusion coefficients of mass fraction, which is set

to the constant value of 6.7e−05 m2/s based on the literature (Fuller

et al., 1966) The SGS Schmidt number Sc SGSwas set to 1.0

3.2 RANS

The unsteady Reynolds-averaged equations for momentum and

mass fraction transports are described as

+

p

u x

u

i

i j j

' '

(10)

Y

k

'

(11) The brackets [ ] denots Reynolds-averaging operation, and the angle

brakets < > is expressed the Favre density averaging The terms with a

fluctuation component expressed with the prime mark are modeled

with a simple gradient diffusion hypothesis as

x

u

3

j j

' '

(12)

x

µ Sc

Y x

i t t k i

'

(13)

where μ t and D t are the turbulent viscosity and the diffusion coefficient,

respectively The turbulent Schmidt number Sc t is the ratio of μ t and D t

μ t is calculated with the formulation according to the standard k–ε

model (Launder and Spalding, 1974) as

=

C μ is the model constant generally set to the value of 0.09 The RANS model requires transport equations for the turbulent kinetic

en-ergy k and its rate of dissipation ε to estimate the value of μ t:

k t

u k

x

[ ]i

t

+

t

u x

µ x

[ ]

i i

i

t i

(16)

where σ k , σε, Cε1, Cε2, and Cε3 are model constants Table 2 summarizes these model constants (Launder and Spalding, 1974; Viollet, 1987) P k

and G k are the production terms of the turbulent kinetic energy by shear stress and buoyancy force, respectively

=

x

j

' '

(17)

=

To close the equation system, P k is modeled with the Eq (12) G k, which is one of the most important factors in this study, is simply ex-pressed as

=

Sc

g x

t i

3.2.1 Dynamic modeling for the turbulent Schmidt number Sc t The Sc tvalue is generally set to the constant value of less than unity (Ishay et al., 2015; Tominaga and Stathopoulos, 2007) Some studies on

ocean engineering mentioned that the Sc t value gradually increases with the increasing stratification strength Venayagamoorthy and Stretch (2010) proposed the following formulation:

Sc C

Ri C

to

g

where Sc t0 is the turbulent Schmidt number under the neutral condition usually set to less than unity The stratification strength is characterized

by the gradient Richardson number Ri gderived with the square ratios of

the Brunt–Väisälä frequency Nu = g

z and the mean shear flow

rate S = ( u + )

x u x

1 2

j j

i in this paper This formulation was de-veloped in terms of scalar time scale ratio T T L/ ; = T L k/ is the

turbu-lent kinetic energy decay time scale, and = T ((1/2)< >' )/2 is the scalar decay time scale, where is the dissipation ratio of scalar fluc-tuation The DNS results of Shih et al (2000) indicated values of

=

C1 1/3and = C2 1/4 However, this formulation was validated with

the DNS data of Ri g = 0 to 0.6, which seems a much smaller range than that in the VIMES experiment

Strang and Fernando (2001) proposed a formulation of Sc t based on the measurement data of the temperature and velocity at the Pacific

Fig 5 Time transient of the vertical distribution of the helium molar fraction

(%) in Case 1in the VIMES experiment

Table 2

Model constants in the standard k–ε model

k 1.00

1.30

C3 0 (in stable layer G k <0), 1 (in unstable layer G k 0) ( Viollet, 1987 )

Ocean near the equator by Peters et al (1988) The proposed

for-mulation indicated that Sc t asymptotes the value of 20 as Ri g ,

although in the above model of Venayagamoorthy and Stretch Sc t, it

becomes the value of infinity when Ri g The formulation of

Ve-nayagamoorthy and Stretch with the threshold value of 20 was applied

in our previous study on the MISTRA HM1-1 test (Abe et al., 2018a,b),

and the CFD simulation predicted well the breakup behavior of the

density stratification The following formulation was applied herein as

the dynamic modeling for Sc t:

Sc C

Ri C

to

g

The minimum value of Sc to was implemented for the neutral and

unstratified conditions

3.3 Simulation case

This section presents the numerical and boundary conditions The

numerical model was validated with respect to the VIMES experiment

To simulate the turbulent behavior in the near-wall region in the

LES, the Van Driest damping function

A

1 exp

d

(22)

was used with = A 25(Van Driest, 1956) That is, the filter width in

Eq (8) was replaced with F d y +is a non-dimensional wall distance

from a wall In the RANS cases, µ t was damped with the following

formulation:

=

>

+

+

+

+

where is the von Karman constant of 0.41 Table 3 summarizes other

details of the LES and RANS

The grid system for the LES was composed of approximately 22.4

million hexahedral elements (Fig 6(a)) The jet inlet face was filled

with 320 surfaces The grid system for the interaction region between

an upward jet and stratification is refined We checked the mesh

suf-ficiency by first performing the simulation on an upward jet with the

injected velocity of 5.0 m/s (Reynolds number Re = W D µ inj 10000) in

the VIMES test vessel The radial profile of the axial mean velocity

W mean at z = 1.2 m (z/D = 40) from the bottom of the test vessel was then compared with the experimental results and literature (Panchapakesan and Lumley, 1993) (Fig 7(a)) The simulated profile was in a good agreement with the experimental data Fig 7(b) shows

the radial profiles of the turbulence fluctuation ( u u[ i i] [ ][ ]u u i i ) in

the vertical component w r.m.s. The overall shape (e.g., flat shape in the jet inner region) was in a good agreement with the experimental data

The axial variation of the axial mean velocity W W inj/ meanalong the jet centerline in the LES was also close to Eq (2) and literature (Panchapakesan and Lumley, 1993) (Fig 8(a)) Moreover, the axial

variation of the normalized turbulence fluctuation w r m s ./W meanalong the jet centerline in the LES was in a good agreement with the data of Panchapakesan and Lumley (1993) exhibiting w r m s ./W mean 0.24 (Fig 8(b)) This comparison indicated that the general jet behavior was reasonably simulated Fig 9(a) shows the axial velocity spectrum at

z = 1.2 m (z/D = 40) from the bottom of the test vessel in the LES The spectra were normalized with the Kolmogorov length scale

= ( / )

k 3 1/4, where =2 s s ij ij and are the dissipation ratio and the kinematic viscosity, respectively, and the Kolmogorov velocity

scale = u k ( )1/4 The figure confirms the region corresponding to the slope of 5/3 shown in a dashed line The extent of the −5/3 range generally increases with the Taylor Reynolds number Re = k r m s w ., which is approximately 330 in this case Compared with the previous study of Saddoughi and Veeravalli (1994), who assembled data of several flows for Re =23 3180, the −5/3 range in this simulation was considered reasonable In other words, the energy cascade by in-ertial transfer was adequately simulated Fig 9(b) also provides the compensated velocity spectrum defined as 2/3 5/3E ( ) In the inertial subrange (−5/3 range), the spectra should be independent of the wa-venumber and equal to the Kolmogorov constant of approximately 0.491 The smoothed data shown by the red solid line was close to this value, although the plots largely fluctuated This result shows that the numerical mesh was sufficient for simulating the turbulent jet behavior The simulation on the stratification breakup by a vertical jet showed no clear criterion for determining whether the numerical mesh was suffi-cient for simulating the interaction behavior Therefore, in this study, the µ SGSvalue was used as the criteria for mesh sufficiency The result for Case 1 showed that the maximum value of µ SGSat the interaction region between the vertical jet and stratification achieved approxi-mately 2.9e−06 Pa⋅s, corresponding to approxiapproxi-mately 16% of the molecular viscosity We consider this value to be small enough for

Table 3

Numerical and boundary conditions of the simulations

Numerical

Space discretization 2 nd -order central difference

TVD (Total variation diminishing) scheme for advection terms Time discretization Euler-implicit

Time marching PIMPLE, Combination of PISO(Pressure Implicit with Splitting of Operator) and SIMPLE (Semi-Implicit Method for Pressure Linked Equations)

Boundary

Turbulent kinetic energy (k) in the

Turbulent dissipation rate (ε) in the

Turbulent kinetic energy (k) in the

Turbulent dissipation rate (ε) in the

yp

3/4 3/2

p is the value of turbulence dissipation rate at the nearest cell center to wally p and k pmean distance from the wall to the cell center and turbulent kinetic energy at the cell center, respectively

simulating the flow and mass transport behavior using the LES

meth-odology

Regarding the numerical mesh for the RANS, we confirmed the

mesh convergency by comparing three different resolutions (Fig 6(b))

Mesh_01 was composed of approximately 0.31 million hexahedral

ele-ments The inlet face was filled with 48 surfaces The region above the

inlet boundary was refined to suppress the excessive flow spreading

The jet impingement region to the stratification was refined in Mesh_02,

which was composed of approximately 0.46 million elements In

Mesh_03 with approximately 0.88 million elements, the overall simu-lation domain was refined from Mesh_02 First, we performed the si-mulation on an upward jet with an injected velocity of 5.0 m/s Fig 10(a) compares the radial profiles of the axial mean velocity W mean

at z = 1.2 m (z/D = 40) among the RANS results using the three re-solutions Furthermore, Figs 7 and 8 compare the axial mean velocity

W mean and turbulence fluctuation w r.m.s on the upward jet in the case with Mesh_02 with the LES and experimental results The vertical jet was reasonably simulated Second, the convergency on the time

Fig 6 Numerical meshes: (a) overall domain in the LES (1: jet injection region; 2: interaction region); and (b) overall domain in RANS (1: coarse mesh with 0.31

million elements; 2: medium mesh with 0.46 million elements; 3: fine mesh with 0.88 million elements)

transient of the helium fraction in Case 1 was confirmed (Fig 10(b))

These figures do not indicate dependence on the numerical mesh The

RANS results with Mesh_02 are shown herein

4 CFD result

Table 1 summarizes the simulation case in this study The LES was

performed only for Case 1 The calculated period was limited only to

62 s from the start of the upward jet injection to save computational

cost and time The statistical processing was performed with the data of

30 s to 62 s, which corresponded to the period of the PIV measurement

in the VIMES experiment For RANS in Case 1, the constant value of Sc t

was set to 0.85 and 8.5 The first value was decided based on the

pre-vious studies that simulated the same kind of stratification erosion

process by several jet types (Kelm et al., 2019, Studer et al., 2018, and

Ishay et al., 2015) The second one was set to validate the effect of the

forcible suppression of the turbulent mixing The Sc t value for the other

cases was set only to 0.85 The dynamic modeling for Sc tgiven by Eq (21) with Sc t0 =0.85was applied in all experimental cases

4.1 Flow and gas concentration fields in the interaction region in case 1

Fig 11 shows the LES-simulated instantaneous flow field in the interaction region between the upward jet and stratification at 46 s in Case 1 Two interesting flow patterns were seen: a large meandering flow and a small eddy structure at the upper part (z > 1.3 m, Fig 11(b)) The first behavior is presumed to result from the jet rapid deceleration and rebounding flow At the upper part of the impinge-ment region (z > 1.3 m), the upward jet was changed to the flow direction horizontally, and a large velocity gradient existed across the

Fig 7 Radial profiles of the (a) axial mean velocity W mean (m/s) and (b) turbulence fluctuation of the axial velocity component w r.m.s (m/s) at z = 1.2 m (z/D = 40) from the bottom of the test vessel The error bars are the standard deviations from five independent measurements

Fig 8 Axial variations of (a) the axial mean velocity W inj /W mean and (b) turbulence fluctuation W r.m.s. /W mean at the jet centerline

density interface Therefore, the small eddy structure was generated

We consider that the LES simulated a reasonable flow behavior in the

interaction region

The time-averaged flow field in the LES showed that the upward jet

arrived at approximately z = 1.3 m (Fig 12(b)) The penetration depth

was close to that in the VIMES experiment Furthermore, some

im-portant flow patterns (i.e., upward jet spreading and magnitudes of the

upward jet and rebounding flow) were similar to those of the

experi-mental result The general flow patterns in all RANS results were in

accordance with the experimental result (Fig 12(c) to (e)), although a

slight difference was found among the RANS results (i.e., the upward jet

arrival point in the case with Sc t = 0.85 was higher than those in the

other cases, while that in the case with Sc t = 8.5 was lower than the

others) This small difference influenced the stratification erosive

pro-cess

The spatial gradient of the helium gas concentration at the inter-action region is very important because the turbulence transport quantity and the production term G k are modeled with Eqs (13) and (19), respectively Fig 13 shows the time-averaged gradient of the helium mass fraction in the vertical direction along the jet centerline from the LES and RANS results The stratification interface was pushed

up (Fig 12(b) to (e)); hence, the gradient became steep The vertical

distribution in the RANS result with the dynamic modeling for Sc t was similar to that of the LES result

Regarding the turbulence fluctuation in the vertical component

w r.m.s., the LES result indicated a similar spatial distribution to the VIMES result (Fig 14(a), (b), and (f)) These results imply the gradual

decay of w r.m.s above the jet impingement region (z > 1.3 m) In

contrast, RANS in the case with Sc t = 0.85 predicted the rapid decay in this region (Fig 14(c) and (f)) The discrepancy was from the turbu-lence production term shown by Eq (17), (18), and (19) Fig 15(a)

shows the vertical profiles of the P k and G k terms along the jet center-line The negative value in this figure means the damping on the tur-bulence kinetic energy, while the positive value denotes the activation

The magnitude of G k in the RANS with Sc t = 0.85 was larger than that

in the LES, and the peak value of the P k term was lower In other words, the mechanism of the turbulence production was not adequately

si-mulated In the case with Sc t = 8.5, the turbulence fluctuation in the upper part gradually decreased This result was seemingly close to the VIMES experimental and LES results (Fig 14(d) and (f)) Moreover,

RANS with the dynamic modeling for Sc t predicted the gradual decay behavior (Fig 14(e) and (f)) Focusing on the turbulence production in

the interaction region, the G k profile in the case with Sc t = 8.5 was

smaller than that in the LES, and the P k profile was larger (Fig 15(a))

Meanwhile, the profiles of G k , P k, and the total balance of the turbulent

kinetic energy production G k + P k predicted by using the dynamic

modeling for Sc t was very close to that in the LES Figs 16 and 17 show

the radial profiles of G k and P k The heights were selected based on the peak value at the jet centerline These figures reveal the good

perfor-mance of the dynamic modeling for Sc t Both distributions were similar

to those in the LES Fig 15(c), Fig 16(b), and Fig 17(b) show the

vertical and radial profiles of Sc t to clarify the effect of the dynamic

modeling in detail The Sc t at the inner jet region (z < 1.3, r < 0.1 m) was a constant value of approximately 0.85 to 2 In the stratification

and side of the vertical jet, the Sc t value gradually increased,

demon-strating that the change of the Sc t value plays a key role in predicting the turbulence properties in the density stratification

4.2 Time transient of the helium molar fraction in case 1

In all RANS, the time transients of the helium molar fraction were qualitatively predicted well (Fig 18) In the lower part of the initial stratification, the molar fraction immediately decreased after the start

of the vertical jet injection This rapid decay was induced by the jet

Fig 9 (a) Velocity spectra of the axial velocity component at z = 1.2 m (z/

D = 40) from the bottom of the test vessel in the LES kis the Kolmogorov

length scale u kis the Kolomogorov velocity scale The solid line is from the LES,

and (b) Compensated velocity spectra of the axial velocity component at

z = 1.2 m (z/D = 40) from the bottom of the test vessel Black circle is the

compensate velocity spectra in the LES, and the red line is the smoothed data

Fig 10 Comparison of the RANS results using three numerical resolutions: (a) radial profile of the axial velocity component W mean (m/s) at 1.2 m (z/D = 40) and (b) time transient of the helium molar fraction (%) in Case 1

impingement into the stratification The small difference at z = 1.3 m

among the RANS results was produced by the change of the jet

pene-tration depth (Fig 12) The decay of the helium molar fraction in the

upper part of the stratification was also qualitatively predicted (i.e.,

slow erosive process before the jet achievement and the rapid decrease

induced by the strong turbulence mixing) Quantitatively, the RANS

result with the constant value of Sc t = 0.85 showed a faster breakup

transient, indicating that the turbulence mixing was overpredicted In

the case with the constant value of Sc t = 8.5, the turbulence mixing in

the jet impingement region was forcibly suppressed, and the time

transients were closer to the experimental data In the case with the

dynamic modeling for Sc t, the time transients of the helium molar

fraction were in a good agreement with the experimental result,

in-dicating that the turbulence mixing in the impingement region was

accurately simulated

5 Discussion

5.1 Turbulence transport phenomena in the interaction region

Hereafter, we focused on the turbulence mixing phenomena with

the spatial visualization of the turbulence helium mass flux obtained

with the CFD simulation Fig 19 shows the turbulent helium mass flux

in the vertical direction [ ' ' ]w Y He and the horizontal integral value of

this turbulent flux In the LES, this turbulent flux was directly derived

from statistical processing The LES result showed negative values in

the interaction region between the jet and stratification This negative

value meant that the helium gas was transported downwardly by the

turbulence mixing Incidentally, the positive value of the turbulence

flux was seen surrounding this region, showing the counter-gradient

diffusion (Komori et al., 1983), which was a buoyancy-driven motion in

the stratified layer (Komori and Nagata, 1996) This behavior

demon-strates that a part of the light gas mixture returned to its original level

In RANS, the turbulent mass flux was modeled by Eq (13) with the

positive diffusion coefficient of µ Sc t/ t; thus, the counter-gradient

dif-fusion was not simulated Focusing on the interaction region, where the

strong turbulence mixing was seen, the simulation result in the case

with Sc t = 0.85 indicates a distribution larger than that in the LES

result The RANS result in the case with Sc t = 8.5 showed that the overall mixing capability seemed lower than that in the case with

Sc t = 0.85 in Fig 19(e) However, the spatial distribution was far from that of the LES result, as shown by the comparison between Fig 19(a) and (c), indicating that the turbulence mixing behavior was different That is, a better agreement, as mentioned in Section 4, was not pro-duced by the reasonable improvement In the case with the dynamic

modeling for Sc t, turbulence mixing was adequately suppressed The

Fig 11 Instantaneous flow field obtained by the LES in Case 1: (a) overall interaction region between the vertical jet and stratification and (b) focusing on the small

eddy structure at the stratification interface The color contour shows the velocity magnitude based on rdial and vertical components u r2+ w2(m/s)

Fig 12 Averaged velocity field in Case 1: (a) VIMES, averaged flow field with line contour of the helium molar fraction in Case 1; (b) LES; (c) RANS (Sc t = 0.85); (d)

RANS (Sc t = 8.5); and (e) RANS (dynamic Sc t) The color contour shows the velocity magnitude based on rdial and vertical components u r2+ w2(m/s)

Fig 13 Time-averaged gradient of the helium mass fraction in the vertical

direction (m−1) along the jet centerline in Case 1 obtained from the LES and RANS results