Horizontal buoyant jet is a fundamental flow regime for hydrogen safety analysis in power industry. The purpose of this studyis to develop a fast non-Boussinesq engineering model the horizontal buoyant round jets. Verification of this integral modelis established with available experimental data and comparisons over a large range of density variations with the CFD codesGASFLOW. The model has proved to be an efficient engineering tool for predicting horizontal strongly buoyant round jets.
Turbulent buoyant jet is a fundamental flow regime inhydrogen safety analysis since it affects hydrogen distributionand mitigation measures when accidents occur [1]. Recentworks on the numerical simulation of helium and steamjet release experiments in the German THAI facility haveshown that the Helium distribution (simulating hydrogen)may depend sensitively on the jet modeling [2]. A systematicinvestigation was therefore started on numerical modeling ofhorizontal turbulent buoyant jets [3].
Few experimental data and calculations on horizontalturbulent buoyant jet with large density variation can befound in the open literatures. Most of the experiments werecarried out for the small density variation when the Boussi-nesq approximation is valid. Pantokratoras [4] modified theintegral Fan-Brooks model [5] to calculate the horizontalpenetration of inclined thermal buoyant water jets, andthe modified model predictions are in a good agreementwith the trajectory measurements. Jirka [6, 7] formulatedintegral model, namely, CorJet for turbulent buoyant jets(round and planar) in unbounded stratified flows, that is,the pure jet, the pure plume, the pure wake. Guerra [8]simulated horizontal buoyant jets (64% helium into airwith various inlet velocity) using CFD code FLOVENT,but unfortunately his work is only from calculation resultsand he did not provide any experiment data for this case.
Swain [9, 10] studied horizontal round hydrogen jets anddetermined the maximum horizontal distance at which ahorizontal hydrogen leak can be ignited. Houf and Schefer[11] developed a model to study the small leaks of hydrogenin the Froude number range where both buoyant and inertialforces are important.
In this study non-Boussinesq integral model for hor-izontal buoyant round jet was derived with the modifiedentrainment hypothesis. The system of conservation equa-tions of the integral model was solved by a forth order Runge-Kutta method to obtain numerical solutions in the transitionregion from jet-like to plume-like.
The problem description and modeling efforts are pre-sented in Sections 2 and 3. Sections 4 and 5 documents thevalidation and modeling uncertainties of the integral model.The concluding remarks are given in Section 6.
2. Problem Description
The horizontal buoyant jet formed from a round orifice isdischarged into the unbounded stagnant uniform ambient,as shown in Figure 1. The density of the ambient is ρa. Theaxis of the jet is taken as a parametrical coordinate s, and thecoordinate n is taken to be normal to the axis s. θ is the angleof s-axis with the horizontal direction. The initial density,velocity, and diameter in the orifice are ρ0,U0, r0. The densityand velocity along the s-axis are ρs , us.
2 Science and Technology of Nuclear Installations
3. Formulation of Horizontal Buoyant Jets
3.1. Basic Governing Equations. In this study, the pressureacross the flow is assumed to be uniform and equal to theambient pressure outside of the boundary. The basic gov-erning equations (neglecting the dissipation and turbulenttransport in comparison with the mean flow) consist ofmass, momentum, energy, and concentration conservationequations,
∇ ·(ρ⇀u)= 0,
∂(ρuui
)
∂x= 0,
∂(ρuuj
)
∂y= Δρg,
∇ ·(ρ⇀u h
)= 0,
∇ ·(ρ⇀u φm
)= 0,
(1)
where h is the specific enthalpy and Φm is the massconcentration.
The divergence theorem is applied, and the basic govern-ing equations become
d
ds
∫ 2π
0
∫∞0
(ρu)r dr dϕ = 2παj−pbρaus = Em,
d
ds
∫ 2π
0
∫∞0
(ρuu cos θ
)r dr dϕ = 0,
d
ds
∫ 2π
0
∫∞0
(ρuu sin θ
)r dr dϕ =
∫ 2π
0
∫∞0
(ρa − ρ
)gr dr dϕ,
d
ds
∫ 2π
0
∫∞0
[ρu(CPT − CPaTa)
]r dr dϕ = 0,
d
ds
∫ 2π
0
∫∞0
(ρuφm
)r dr dϕ = 0,
dx
ds= cos θ,
dy
ds= sin θ,
(2)
where Em is the local mass entrainment rate and αj−p is thelocal mass entrainment coefficient.
A system of first-order ordinary differential equa-tions was obtained after the integration, where the sevenunknowns are the density, velocity, temperature along thetrajectory, ρs, us, Ts, the characteristic jet width b, the localangle of the jet with respect to the horizontal axis θ, andthe local coordinates of the jet trajectory x, y. With initialconditions, the system of ordinary differential equations wassolved by aforth order Runga-Kutta method to obtain thehorizontal buoyant jet trajectory, the velocity, the density, thetemperature and the tracer concentration.
ρ0,U0, r0
ρs,us
Axis
r
s
θ
ρa>ρ0g
x
Figure 1: Definition diagram for horizontal buoyant jet dischargesfrom round orifice into the unstratified ambient.
3.2. Basic Assumptions. The general assumptions made inthis investigation are as follows.
(1) The flow is fully turbulent which means there is noReynold number dependence.
(2) The profiles of velocity, density, and temperatureare similar at all cross-sections normal to the jettrajectory.
(3) Longitudinal turbulent transport is small comparedwith latitudinal convective transport.
3.3. Profiles of Velocity, Density, and Temperature. Velocityprofile is assumed to be Gaussian distribution:
u = use−r2/b2
, (3)
where b is a characteristic jet width.Density deficiency profile with respect to the ambient
density in a uniform ambient is assumed to be Gaussian:
ρa − ρρa
=(ρa − ρsρa
)e−r
2/(λb)2
, (4)
where λb is the characteristic length of the profiles; λ2 is theturbulent Schmidt number, which is assumed to be constantand is usually found to be somewhat larger than 1 for smalldensity ratio cases. In this study λ is 1.2. It should be notedthat λ2 is not necessarily constant as the buoyant jet evolvesfrom jet-like to plume-like.
Due to the large density or temperature variationbetween the jet and the ambient considered in the non-Boussinesq model, the density in the trajectory ρs is thefunction of temperature Ts and mass faction Φms. Therefore,the energy equation should be solved. It is well knownthat the cross-sectional profiles of normalized mean axialvelocity and concentration are Gaussian-like in the zoneof established flow (ZEF). In this study, the profile oftemperature is also assumed to be Gaussian distribution:
Ta − TTa
=(Ta − TsTa
)e−r
2/(λb)2
, (5)
Science and Technology of Nuclear Installations 3
3.4. Entrainment Hypothesis. To close the equations system,the mass entrainment rate should be specified. The entrain-ment relation for the horizontal round jet is given by:
Em = 2πbρaue = 2παj−pbρaus, (6)
where Em is the local mass entrainment rate, ue is the localentrainment velocity, us is the characteristic velocity along s-axis, ρs is the local density along s-axis, ρa is the density ofambient, and αj−p is the local mass entrainment coefficient .
For low-momentum buoyant jets, experimental dataindicates that the local rate of entrainment increases as thejets leaves the momentum-dominated region and enters aregion where the effects of buoyancy become more pro-nounced. In Jirka’s paper [6], the local rate of entrainmentis specified as
Em = 2πbUs
(αj + αp
Fr2p
Fr2s
sin θ
). (7)
List summarized much of the work on the entrainmenthypothesis and proposed values of αj = 0.052 ± 0.003 forthe pure jets and αp = 0.0833± 0.0042 for the pure plumes[12]. To model the entrainment in the region of transitionfrom jet-like to plume-like, an empirical function for thevertical plane jet proposed,
αj−p = αj +(αp − αj
)(Rij−pRip
)2
, (8)
Rij−p is the local Richardson number in the trajectory whichis defined as
Rij−p =mβ1/2
mo5/4, (9)
where m is the mass flux, mo is the momentum flux, β isthe local buoyancy flux. The Richardson number in the pureplume region Rip is a constant.
In the non-Boussinesq model, the effect of large densityvariation should be considered in the entrainment coeffi-cient. In this study the local entrainment coefficient for thehorizontal buoyant jet is assumed as:
αj−p =⎡⎣αj +
(αp − αj
)(Rij−pRip
)2
sin θ
⎤⎦∗
(ρsρa
)1/2
(10)
for the round jet, the exponent is 1/2, αj = 0.055 for thepure jets and αp = 0.085 for the pure plume. We should notethat the value of αj here is obtained from the flow with lowvelocity. For the flow with high velocity, αj may be varied. Itwill be discussed in the section of modeling uncertainties.
4. Validations
4.1. Pure Jet. For the pure jet (θ = 0◦, ρa = ρ0), there is nobuoyancy acting on the flow and the jet is only dominated by
10−210−1100
s/D0
Albertson (1950)Crow and Charnpagne (1971)Capp (1983)Labus and Symons (1971)Hussein (1994)Papanicolaou and List (1988)Analytical solutionNB integral model
10−2
10−1
100
Us/U
0
Figure 2: Centerline velocity decay for pure jets.
the momentum. The conservation equations of the pure jetcan be expressed as:
d
ds
(πρausb
2s
) = 2παjetρausbs,
d
ds
(π
2ρau
2s b
2s
)= 0,
d
ds
(πλ2
1 + λ2csusb
2s
)= 0.
(11)
The ratio of us and the initial velocity U0 is,
usU0
=√
24αj
(s
D0
)−1
. (12)
The centerline dilution is,
1 + λ2
λ2Sc =
(√8παj
) s
D0. (13)
The centerline velocity decay and concentration dilution areshown in Figures 2 and 3. When αjet and λ are 0.055 and 1.2,the results of the non-Boussiensq integral model developedin this study have good agreement with the experimental dataand the analytical solutions in (12) and (13).
4.2. Bouyant Jet with Small Density Variation. The horizontalbuoyant jets with small density variations (<5–10%) arepredicted with the non-Boussinesq integral model. Thenitrogen (300 K, 1 bar) is injected horizontally into theambient (300 K, 1 bar) with various initial Froude numbers.Figure 4 shows the normalized vertical trajectory Y/LM as afunction of X/LM for a variety of experimental conditions
4 Science and Technology of Nuclear Installations
100101
s/D0
Albertson (1950)Crow and Charnpagne (1971)Hill (1972)Labus and Symons (1971)Ricou and Spalding (1961)Papanicolaou and List (1988)Analytical solution & CorJet model, Jirka (2004)NB integral model
1
10
100
1.7Sc
Figure 3: Concentration decay along the centerline for pure jets.
and the comparisons of the non-Boussinesq integral modeland the Boussinesq-based CorJet model.
The length scale LM for the transition of horizontalbuoyant round jet to plume is defined as
LM = mo3/40
j1/20, (14)
where
mo0 = U20A0, j0 = U0
((ρa − ρ0
)g
ρ0
)A0. (15)
The normalized centerline dilutionSc/Fr0 as a function ofY/LM is depicted in Figure 5. The Froude number is:
Fr0 = U0√((ρa − ρ0
)g/ρ0
)R0
. (16)
The predictions of non-Boussinsq model agree well withthe experimental data. When the initial Froude number Fr0 ishigh enough the non-Boussinesq model’s results will collapseto the CorJet model. It indicates that in high Froude numberflow the influence of small density variation can be neglected.However in the flows with low Froude numbers it seemsthat the effects of the buoyancy should be considered. Thenormalized trajectory will not collapse when the Froudenumber is small which means the Boussinesq approximationis invalid when the buoyancy effect is comparative with themomentum effect.
The non-Boussinesq integral model provides a satis-factory transition behavior for the horizontal buoyant jetswith small density variations from the jet-like to plume-like region. Figures 6, 7, and 8 depict the predictions
543210
X/LM
Fan (1967) Fr0 = 14–42CorJet model, Jirka (2004)Fr0 = 10Fr0 = 20Fr0 = 30Davidson and Pun (2000) Fr0 = 10–25
Fr0 = 50Fr0 = 100Fr0 = 300Fr0 = 500Fr0 = 1000
0
1
2
3
4
5
6
7
Y/LM
Figure 4: Normalized trajectories of horizontal buoyant jet.
Figure 5: Normalized centerline dilutions of horizontal buoyant jet.
of non-Boussineq integral model: the decays of velocities,Froude numbers, and the entrainment coefficients along thecenterline of the horizontal buoyant jets.
4.3. Buoyant Jet with Large Density Variation. The horizontalbuoyant jets with large density variations, for instancehydrogen or helium injecting into air, have not receivedsufficient research before, and almost no experimental datacould be found in the open literature. CFD code GASFLOW[13, 14] was used to validate the non-Boussinesq integralmodel. Helium at 300 K, 1 bar is injected into the air (300 K,1 bar) with various efflux velocities. The diameter of theorifice is 0.0845 m.
Science and Technology of Nuclear Installations 5
104103102101100
s/D0
Fr0 = 10Fr0 = 20Fr0 = 30Fr0 = 50
Fr0 = 100Fr0 = 300Fr0 = 500Fr0 = 1000
10−3
10−2
10−1
100
Us/U
0
Figure 6: Velocity decay of horizontal buoyant jet.
10510410310210110010−1
s/D0
Fr0 = 10Fr0 = 20Fr0 = 30Fr0 = 50
Fr0 = 100Fr0 = 300Fr0 = 500Fr0 = 1000
100
101
102
103
Fr
Figure 7: Froude number decay of horizontal buoyant jet.
Figure 9 shows the trajectories of the horizontal buoyantjets. The non-Boussinesq integral model also provides goodtransition behavior for the horizontal buoyant jets with largedensity variations. The decays of the velocity and concen-tration along the trajectory are shown in Figures 10 and11. Although small discrepancies exist between the resultsof the integral model and CFD code GASFLOW, the non-Boussinesq integral model is capable of predicting the hori-zontal buoyant jets with large density variations. Experimen-tal data are expected for the further validation of the model.
5. Modeling Uncertainties
Before the Gaussian profiles are reached, the initialunsheared profiles undergo changes in form of peripherally
10510410310210110010−1
s/D0
Fr0 = 10Fr0 = 20Fr0 = 30Fr0 = 50
Fr0 = 100Fr0 = 300Fr0 = 500Fr0 = 1000
0.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
En
trai
nm
ent
coeffi
cien
tα
Figure 8: Entrainment coefficients of horizontal buoyant jet.
32.521.510.50
X (m)
NB integral model (10 m/s)NB integral model (20 m/s)NB integral model (30 m/s)
GASFLOW (10 m/s)GASFLOW (20 m/s)GASFLOW (30 m/s)
0
0.5
1
1.5
2
2.5
3Y
(m)
Figure 9: Trajectories of horizontal buoyant jet with large densityvariations.
growing axis symmetric mixing layers. This initial region iscalled the zone of flow establishment which lacks of self-similarity. The transition in this region is complex and rapid,and the distance is up to 5–10 diameter of the orifice. Adistance of 5–10 diameters from the orifice is shifted in thestudy.
The mass entrainment coefficients in this study wereobtained under the experimental conditions when thevelocity and density variation are not so high. How the highvelocity and large density variation affect the entrainmentcoefficient is not clear. In the recent simulation of theunderexpanded hydrogen jet [15], it indicates that theentrainment coefficient for the pure jet, αj , increases from0.055 to 0.065–0.07 when the velocity is sonic or supersonic.
6 Science and Technology of Nuclear Installations
32.521.510.50
Y (m)
NB integral model (10 m/s)NB integral model (20 m/s)NB integral model (30 m/s)
GASFLOW (10 m/s)GASFLOW (20 m/s)GASFLOW (30 m/s)
0
3
6
9
12
15
Us
(m/s
)
Figure 10: Velocity decay of horizontal buoyant jet with largedensity variations.
3210
Y (m)
NB integral model (10 m/s)NB integral model (20 m/s)NB integral model (30 m/s)
GASFLOW (10 m/s)GASFLOW (20 m/s)GASFLOW (30 m/s)
0
0.2
0.4
0.6
0.8
1
Con
cen
trat
ion
,Cs
Figure 11: Concentration decay of horizontal buoyant jet with largedensity variations.
The mechanisms of these uncertainties needs furtherstudy in the future work.
6. Concluding Remarks
This non-Boussinesq integral model developed in the studyis a fast engineering model to solve the horizontal buoyantround jets problems. The model was validated by the pure jet,horizontal buoyant jets with small/large density variations,and good agreements with the experimental data, and CFDpredictions were obtained.
For strongly buoyant jet the Boussinesq approximationis violated which will over-predict the mass entrainmentand under-estimate the buoyancy effect [16]. It reveals that
the Boussinesq approximation is valid when the densityvariation is less than 10%.
The entrainment assumption is a key requirement forthe integral model. The entrainment assumption taking intoaccount the Richard number and the angle θ was used in theintegral model, and satisfactory predictions for the modelingof horizontal buoyant jets were provided. The effects oflarge density variation and high velocity on the entrainmentcoefficient need further study in the future.
References
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[2] P. Royl, U. J. Lee, J. R. Travis, and W. Breitung, “Benchmarkof the 3D CFD code GADFLOW II with containmentthermal hydraulic tests from HDR and THAI,” in CFD4NRSConference, Munich, Germany, September 2006.
[3] J. Xiao, J. Travis, and W. Breitung, “Numerical simulation offree horizontal turbulent buoyant planar jets and plumes,”in Proceedings of the Annual Meeting on Nuclear Technology,Karlsruhe, Germany, May 2007.
[4] A. Pantokratoras, “Horizontal penetration of inclined thermalbuoyant water jets,” International Communications in Heat andMass Transfer, vol. 25, no. 4, pp. 561–569, 1998.
[5] L. N. Fan and N. H. Brooks, Numerical Solution of TurbulentBuoyant Jets Problems, California Institute of Technology,Pasadena, Calif, USA, 1969.
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[9] M. Swain, “Codes and standards analysis, 2004 annualprogram review meeting of the hydrogen,” Fuel cell &infrastructure program of the US DOE, 2004.
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[11] W. Houf and R. Schefer, “Small-scale unintended releases ofhydrogen,” in Proceedings of the Annual Hydrogen Conferenceand Hydrogen Expo, San Antonio, Tex, USA, March 2007.
[12] W. Rodi, Turbulent Buoyant Jets and Plumes, vol. 6 of HMT TheScience & Applications of Heat and Mass Transfer, PergamonPress, Oxford, UK, 1982.
[13] J. R. Travis, J. W. Spore, P. Royl, et al., GASFLOW: AComputational Fluid Dynamics Code for Gases, Aerosols andCombustion. Volume 1: Theory and Computational Model, FzKand LANL, 2001.
[14] J. R. Travis, J. W. Spore, P. Royl, et al., GASFLOW: AComputational Fluid Dynamics Code for Gases, Aerosols andCombustion. Volume 2: User’s Manual, FzK and LANL, 2001.
[15] A. Veser, G. Stern, M. Schwall, et al., “Structure and flamepropagation regimes in turbulent hydrogen jets,” in Proceed-ings of 7th International Symposium on Hazards, Prevention,and Mitigation of Industrial Explosions, vol. 1, pp. 198–207,Saint-Petersburg, Russia, July 2008.
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[16] J. Xiao, J. Travis, and W. Breitung, “Non-boussinesq integralmodel for horizontal turbulent strongly buoyant plane jets,”in Proceedings of the 16th International Conference on NuclearEngineering, Orlando, Fla, USA, May 2008.
