Basic Concepts about CFD Models - unipi.it · 2011. 1. 13. · CFD codes , mostly making use of...

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Basic Concepts about CFD ModelsBasic Concepts about CFD Models

Walter AmbrosiniWalter Ambrosini

Associate Professor in Associate Professor in NuclearNuclear PlantsPlants

at the at the UniversityUniversity ofof PisaPisa

Lappeenranta University of TechnologyLappeenranta University of Technology

Summer School in Heat and Mass TransferSummer School in Heat and Mass TransferAugust 18 August 18 –– 20, 201020, 2010

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SummarySummary

�� General remarks on turbulent flowGeneral remarks on turbulent flow

–– Instability of laminar flowInstability of laminar flow

–– Statistical treatment of turbulent flowStatistical treatment of turbulent flow

–– Momentum transfer in turbulent flowMomentum transfer in turbulent flow

–– Heat transfer in turbulent flowHeat transfer in turbulent flow

�� Basic concepts about computational modelling of turbulent flowsBasic concepts about computational modelling of turbulent flows

–– Length scales in turbulenceLength scales in turbulence

–– Direct Numerical Simulation (DNS)Direct Numerical Simulation (DNS)

–– Large Eddy Simulation (LES)Large Eddy Simulation (LES)

–– Reynolds Averaged Reynolds Averaged NavierNavier--Stokes equations (RANS)Stokes equations (RANS)

�� TwoTwo--phase flow applicationsphase flow applications

�� Prediction of heat transfer deterioration Prediction of heat transfer deterioration

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General remarks on turbulent flowGeneral remarks on turbulent flowInstability of Laminar Flow Instability of Laminar Flow -- 11

•• The transition from laminar flow to turbulence is The transition from laminar flow to turbulence is an example of an example of

flow instabilityflow instability::

→→ beyond a certain threshold, beyond a certain threshold, inertia overcomes viscous inertia overcomes viscous

forcesforces and the motion cannot be anymore orderedand the motion cannot be anymore ordered

→→ this was shown by this was shown by Osborne ReynoldsOsborne Reynolds in a classical in a classical

experimentexperiment

44

•• This transition occurs in many different systems:This transition occurs in many different systems:

→→ pipe flowpipe flow

→→ boundary layersboundary layers


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→→ free jetsfree jets

→→ wakeswakes


66

•• In order to study stability of a nonlinear system by analytical In order to study stability of a nonlinear system by analytical

means the methodology of means the methodology of linear stability analysislinear stability analysis is often is often

adoptedadopted

•• This has the objective to determine This has the objective to determine the stability conditions the stability conditions

consequent to infinitesimal perturbationsconsequent to infinitesimal perturbations: e.g., for a 2D : e.g., for a 2D

boundary layer it isboundary layer it is


EXAMPLES OF TRANSIENT EXAMPLES OF TRANSIENT

ANALYSESANALYSES

CavityCavity

RB ConvectionRB Convection

Buoyant JetBuoyant Jet

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•• Turbulence introduces a large degree of Turbulence introduces a large degree of ““sensitivity to initial sensitivity to initial

conditions (SIC)conditions (SIC)”” that is typical of that is typical of ““deterministic chaosdeterministic chaos””

•• By this, it is meant that By this, it is meant that turbulent motion is not turbulent motion is not ““randomrandom””, ,

though it appears fluctuating in a similar manner, though it appears fluctuating in a similar manner, since the since the

equations governing the system are well definedequations governing the system are well defined

•• This characteristic is shared with many different This characteristic is shared with many different ““chaoticchaotic””

systemssystems, even governed by simple equations, even governed by simple equations


dRe

dτ = Gr

Ψ12

- L

D f'(Re) Re |Re|

dΨ1

dτ = π Re Ω1 - π

2 Fo Ψ1 + 4

π sin γ (

d

dΩ1

dτ = - π Re Ψ1 - π

2 Fo Ω1 + 4

π cos γ

Heating

Cooling

γ

88

•• Owing to the fluctuating nature of the turbulent flow field, it Owing to the fluctuating nature of the turbulent flow field, it is is

customary (after Reynolds) customary (after Reynolds) to introduce an appropriate time to introduce an appropriate time

averagingaveraging of any specific value (of any specific value (““intensiveintensive””) of major ) of major

““extensiveextensive”” variablesvariables

•• The attempt is quite evidently to write The attempt is quite evidently to write equations in terms of equations in terms of

time averaged variablestime averaged variables, structurally similar to those of , structurally similar to those of

laminar flowlaminar flow

•• This attempt is successful, but This attempt is successful, but fluctuations cannot be forgottenfluctuations cannot be forgotten

General remarks on turbulent flowGeneral remarks on turbulent flowStatistical Treatment of Turbulent Flow Statistical Treatment of Turbulent Flow -- 11

99

•• In particular, In particular, the following quantities have overwhelming the following quantities have overwhelming

importanceimportance

•• Turbulence intensity is strictly related to the turbulence kinetTurbulence intensity is strictly related to the turbulence kinetic ic

energyenergy

•• This is one of the most important quantities adopted in present This is one of the most important quantities adopted in present

CFD codesCFD codes, mostly making use of , mostly making use of ““twotwo--equation modelsequation models””, to be , to be

described later ondescribed later on


1010

•• The general balance equations in local and instantaneous The general balance equations in local and instantaneous

formulation are then averagedformulation are then averaged making use of the above making use of the above

described averaging operatordescribed averaging operator

•• After simplifications (described in lecture notes), an averaged After simplifications (described in lecture notes), an averaged

form is finally reached showing that the attempt to get equationform is finally reached showing that the attempt to get equations s

similar to those of laminar flow leaves an additional termsimilar to those of laminar flow leaves an additional term

•• This term, having a clear This term, having a clear ““advectiveadvective”” nature, points out that nature, points out that

fluctuations do play a role in transfers: this role represents afluctuations do play a role in transfers: this role represents a

sort of additional sort of additional ““mixingmixing”” due to turbulencedue to turbulence


1111

•• In analogy with the molecular motion, the basic idea is thereforIn analogy with the molecular motion, the basic idea is therefore e

to interpret such term as an to interpret such term as an additional diffusion due to additional diffusion due to

turbulenceturbulence

•• The momentum and energy balance equations contain this term The momentum and energy balance equations contain this term

that calls for a proper modellingthat calls for a proper modelling


1212

•• The The ““Reynolds stress tensorReynolds stress tensor”” appears in momentum equationsappears in momentum equations

•• The Reynolds stresses account for the additional momentum The Reynolds stresses account for the additional momentum

flux due to eddiesflux due to eddies

General remarks on turbulent flowGeneral remarks on turbulent flowMomentum Transfer in Turbulent Flow Momentum Transfer in Turbulent Flow -- 11

1313

•• It is then customary to adopt the It is then customary to adopt the ““BoussinesqBoussinesq approximationapproximation””

based on a definition of based on a definition of ““turbulent momentum diffusivityturbulent momentum diffusivity”” (eddy (eddy

viscosity)viscosity), trying to define a simple constitutive relationship for , trying to define a simple constitutive relationship for

the Reynolds stressthe Reynolds stress

•• The quantityThe quantity ννννννννTT is no more a property of the fluid, but also is no more a property of the fluid, but also

depends on flow. depends on flow.

•• Of course, Of course, the the BoussinesqBoussinesq approximation shifts the toughness approximation shifts the toughness

of the modelling problem to the definition of the eddy viscosityof the modelling problem to the definition of the eddy viscosity


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•• By the way, many different kinds of turbulence can be By the way, many different kinds of turbulence can be

envisaged, ranging from ideally homogeneous and isotropic to envisaged, ranging from ideally homogeneous and isotropic to

more realistically heterogeneous and anisotropicmore realistically heterogeneous and anisotropic

•• Wall turbulenceWall turbulence is a classical example of the latter cases:is a classical example of the latter cases:

•• Eddy viscosity models have therefore the very tough job to Eddy viscosity models have therefore the very tough job to

reintroduce the complexity lost in the simple reintroduce the complexity lost in the simple BoussinesqBoussinesq

approximationapproximation


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•• It is rather instructive and useful to consider It is rather instructive and useful to consider the distribution of the distribution of

velocity close to a plane wallvelocity close to a plane wall; different quantities of widespread ; different quantities of widespread

use in CFD are introduced at this stageuse in CFD are introduced at this stage

•• A A universal logarithmic velocity profileuniversal logarithmic velocity profile is found both on the is found both on the

basis of simple theoretical considerations and experimentsbasis of simple theoretical considerations and experiments


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•• The effect of turbulence in the transport of momentum can be The effect of turbulence in the transport of momentum can be

clearly seen in comparing the distributions of velocity in the clearly seen in comparing the distributions of velocity in the

classical case of a circular pipe for laminar and turbulent flowclassical case of a circular pipe for laminar and turbulent flowss

•• The flatter profile observed in the case of turbulent flow is thThe flatter profile observed in the case of turbulent flow is the e

direct consequence of the direct consequence of the increasing efficiency in momentum increasing efficiency in momentum

transfer far from the walltransfer far from the wall due to the mixing promoted by due to the mixing promoted by

turbulenceturbulence


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•• The The averaged total energy equationaveraged total energy equation and the and the steady thermal steady thermal

energy equation in terms of temperatureenergy equation in terms of temperature can be written ascan be written as

•• Also in these cases additional terms to be modelled appear, e.g.Also in these cases additional terms to be modelled appear, e.g.::

•• The rationale for evaluating the turbulent contribution is similThe rationale for evaluating the turbulent contribution is similar ar

as in the case of momentumas in the case of momentum

where where ααααααααTT is the is the ““turbulent thermal diffusivityturbulent thermal diffusivity””

General remarks on turbulent flowGeneral remarks on turbulent flowHeat Transfer in Turbulent Flow Heat Transfer in Turbulent Flow -- 11

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•• The picture of the turbulent transfer phenomenon is therefore The picture of the turbulent transfer phenomenon is therefore

the same as for momentum: the same as for momentum:

•• The relation between the two turbulent diffusivities of heat andThe relation between the two turbulent diffusivities of heat and

momentum poses an additional problemmomentum poses an additional problem


1919

•• A simple but effective way to establish this relationship is to A simple but effective way to establish this relationship is to

define a constant define a constant ““turbulent turbulent PrandtlPrandtl numbernumber””,, in analogy with in analogy with

the molecular one assuming that, as a consequence of the the molecular one assuming that, as a consequence of the

Reynolds analogy, this could be in the range of unityReynolds analogy, this could be in the range of unity

•• The assumptionThe assumption in this case in this case is that the same coherent is that the same coherent

structures carrying momentum are also responsible of heat structures carrying momentum are also responsible of heat

transfertransfer

•• However, However, this assumption holds acceptably for fluids having this assumption holds acceptably for fluids having

nearly unity molecular nearly unity molecular PrandtlPrandtl numbernumber; in the other cases, ; in the other cases,

different approaches should be useddifferent approaches should be used


2020

•• In turbulent flow an In turbulent flow an ““energy cascadeenergy cascade”” occurs representing the occurs representing the

transfer of turbulence kinetic energy from larger to smaller transfer of turbulence kinetic energy from larger to smaller

eddieseddies

Basic concepts about computational Basic concepts about computational

modelling of turbulent flowsmodelling of turbulent flowsLength Scales in Turbulence Length Scales in Turbulence -- 11

•• As such, turbulence can be As such, turbulence can be

considered as considered as a phenomenon a phenomenon

characterised by a wide range of characterised by a wide range of

lengthslengths at which interesting at which interesting

phenomena do occur:phenomena do occur:

→→ from from the integral length the integral length scalescale, , llllllll, at which energy is , at which energy is

extracted from the mean flowextracted from the mean flow

→→ to to the the KolmogorovKolmogorov length length

scalescale, , ηηηηηηηη, at which turbulence , at which turbulence kinetic energy is finally kinetic energy is finally

dissipated into heatdissipated into heat

2121

•• It must be noted that the It must be noted that the KolmogorovKolmogorov length scale, length scale, ηηηηηηηη,, is small is small

but still large with respect to the molecular but still large with respect to the molecular ““mean free pathmean free path””::

so, turbulence can still be studied so, turbulence can still be studied

basing on the continuum assumptionbasing on the continuum assumption

•• The integral length scale, The integral length scale, llllllll,, characterising large eddies can be characterising large eddies can be

defined as the average length over which a fluctuating defined as the average length over which a fluctuating

component keeps correlated, i.e. the quantity component keeps correlated, i.e. the quantity is not is not

negligible negligible

•• On both dimensional and experimental basis, it can be shown On both dimensional and experimental basis, it can be shown

thatthat

andand

with with ; therefore, ; therefore,



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Basing on these considerations, Basing on these considerations, it can be concluded that:it can be concluded that:

•• an adequate representation of turbulence should an adequate representation of turbulence should take into take into

account the phenomena of production and dissipation of account the phenomena of production and dissipation of

turbulence kinetic energy at the different scalesturbulence kinetic energy at the different scales

•• in this respect, in this respect, two different strategiestwo different strategies can be envisaged:can be envisaged:

→→ simulating the transient evolution of vortices of different simulating the transient evolution of vortices of different

sizessizes, putting a convenient lower bound for the smallest , putting a convenient lower bound for the smallest

scale scale (DNS, LES, DES)(DNS, LES, DES)

→→ simulating turbulence on the basis of the above described simulating turbulence on the basis of the above described

statistical approachstatistical approach, introducing appropriate production , introducing appropriate production

and dissipation terms to approximately represent the and dissipation terms to approximately represent the

effects of the energy cascade effects of the energy cascade (RANS)(RANS)



2323


modelling of turbulent flowsmodelling of turbulent flowsDirect Numerical Simulation (DNS) Direct Numerical Simulation (DNS) -- 11

•• This methodology follows the former of the two mentioned This methodology follows the former of the two mentioned

routes, routes, trying to simulate with the highest possible space and trying to simulate with the highest possible space and

time detail the evolution of vortices of all relevant sizestime detail the evolution of vortices of all relevant sizes

•• The assumption behind this technique is that the The assumption behind this technique is that the NavierNavier--Stokes Stokes

equations are rich enough to describe the turbulent flow equations are rich enough to describe the turbulent flow

behaviour with no need of additional constitutive laws; for behaviour with no need of additional constitutive laws; for

incompressible flow it is:incompressible flow it is:

•• The web is full of fascinating pictures and movies about DNS The web is full of fascinating pictures and movies about DNS

resultsresults

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modelling of turbulent flowsmodelling of turbulent flowsDirect Numerical Simulation (DNS) Direct Numerical Simulation (DNS) -- 22

•• The application of this technique is The application of this technique is very demanding in terms of very demanding in terms of

computational resourcescomputational resources: representing flows of technical : representing flows of technical

interest is very challenging and requires massive parallel interest is very challenging and requires massive parallel

computingcomputing

•• However the technique is very promising and it is However the technique is very promising and it is sometimes sometimes

used to provide data having a similar reliability to experimentsused to provide data having a similar reliability to experiments

with greater detail in local valueswith greater detail in local values

•• In fact, if used with enough detail, DNS can provide data which In fact, if used with enough detail, DNS can provide data which

can be hardly obtained in similar detail with experimentscan be hardly obtained in similar detail with experiments

•• In addition to be an interesting field of research, In addition to be an interesting field of research, DNS is DNS is

therefore used also to provide data on which empirical therefore used also to provide data on which empirical

turbulence model can be validatedturbulence model can be validated

CFDCFD--FigureFigure--1.ppt1.ppt

2525


modelling of turbulent flowsmodelling of turbulent flowsLarge Eddy Simulation (LES) Large Eddy Simulation (LES) -- 11

•• At a more reduced level of detail, At a more reduced level of detail, LES is aimed at simulating LES is aimed at simulating

only larger eddies, while the smaller scales are treated by only larger eddies, while the smaller scales are treated by

subgridsubgrid--scale (SGS) modelsscale (SGS) models

•• In other words, there are In other words, there are two different length scalestwo different length scales::

→→ the large scales that are directly solved as in DNS;the large scales that are directly solved as in DNS;

→→ the smaller scales that are treated by SGS modelsthe smaller scales that are treated by SGS models

•• As such, LES is computationally more efficient than DNS and As such, LES is computationally more efficient than DNS and

may be also relatively accuratemay be also relatively accurate

•• A key point in LES is introducing a spatial filtering for the A key point in LES is introducing a spatial filtering for the

smaller scalessmaller scales

2626



•• The filters can be of different types:The filters can be of different types:

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2828



•• Once the resolvable scales are defined, the averaged NOnce the resolvable scales are defined, the averaged N--S equations S equations

are written in averaged formare written in averaged form

2929



•• The advection term can be manipulated asThe advection term can be manipulated as

or alsoor also

•• Anyway, introducing the Anyway, introducing the subgridsubgrid--scale stresses (or adopting slightly scale stresses (or adopting slightly

different definitions)different definitions)

it can be finally obtainedit can be finally obtained

3030



•• So, So, the fundamental problem is defining the the fundamental problem is defining the subgridsubgrid scale stressesscale stresses

•• In 1963, In 1963, SmagorinskySmagorinsky defined a model based on the following defined a model based on the following

equationsequations

where Cwhere CSS is the is the SmagorinskySmagorinsky coefficient representing a parameter to coefficient representing a parameter to

be adjusted for the particular problem to be dealt with; values be adjusted for the particular problem to be dealt with; values in the in the

range 0.10 to 0.24 have been adopted for typical problemsrange 0.10 to 0.24 have been adopted for typical problems

•• LES LES is presently promising as a design tool, but still heavy from this presently promising as a design tool, but still heavy from the e

computational point of viewcomputational point of view

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modelling of turbulent flowsmodelling of turbulent flowsReynolds Averaged Reynolds Averaged NavierNavier--Stokes (RANS) models Stokes (RANS) models -- 11

•• As already mentioned, the Reynolds averaging process leads to As already mentioned, the Reynolds averaging process leads to

momentum equations in which turbulence is represented by momentum equations in which turbulence is represented by the the

Reynolds stressReynolds stress

•• The The BoussinesqBoussinesq approximation suggests thatapproximation suggests that

•• Moreover if the Reynolds analogy is adopted by specifying a consMoreover if the Reynolds analogy is adopted by specifying a constant tant

turbulent turbulent PrandtlPrandtl number, also the eddy thermal diffusivity is related to number, also the eddy thermal diffusivity is related to

the eddy viscositythe eddy viscosity

•• So,So, the main problem is reduced to specifying the eddy viscositythe main problem is reduced to specifying the eddy viscosity

2 22

3 3

jiij T ij ij T ij

j i

wwS k k

x xτ ρν ρ δ ρν ρ δ

∂∂= − = + − ∂ ∂

3232



•• Models of different complexity can be adoptedModels of different complexity can be adopted in this aim, classified in this aim, classified

on the basis of the number of the additional partial differentiaon the basis of the number of the additional partial differential l

equations to be solved:equations to be solved:

1.1. Algebraic or zeroAlgebraic or zero--equation modelsequation models

2.2. OneOne--equation modelsequation models

3.3. TwoTwo--equation modelsequation models

•• An important distinction between turbulence models is anyway theAn important distinction between turbulence models is anyway the

one between one between complete and incomplete modelscomplete and incomplete models::

�� completenesscompleteness of the model is related to its capability to of the model is related to its capability to

automatically define a characteristic length of turbulenceautomatically define a characteristic length of turbulence

�� in a complete model, therefore, only the initial and boundary in a complete model, therefore, only the initial and boundary

conditions are specifiedconditions are specified, with no need to define case by case , with no need to define case by case

parameters depending on the particular considered flowparameters depending on the particular considered flow

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ALGEBRAIC MODELSALGEBRAIC MODELS

•• Possibly the best known algebraic model is the one obtained by tPossibly the best known algebraic model is the one obtained by the he

mixing length theory of mixing length theory of PrandtlPrandtl (1925)(1925)

where where llllllllmixmix is the mixing length; the model is similar to the one for is the mixing length; the model is similar to the one for

molecular viscositymolecular viscosity in which kinematic viscosity is a interpreted as in which kinematic viscosity is a interpreted as

the product of a mean molecular velocity by a length (the mean fthe product of a mean molecular velocity by a length (the mean free ree

path)path)

•• In the presence of a wall, it is assumed In the presence of a wall, it is assumed where the constant where the constant

must be adjusted on an empirical basismust be adjusted on an empirical basis

•• The mixing length theory has received different reformulations, The mixing length theory has received different reformulations, but but

its character of incompleteness makes models based on transport its character of incompleteness makes models based on transport

equations to be preferableequations to be preferable

3434



PARTIAL DIFFERENTIAL EQUATION MODELSPARTIAL DIFFERENTIAL EQUATION MODELS

•• Referring from here on to the specific Reynolds stress tensorReferring from here on to the specific Reynolds stress tensor

it is possible to derive a it is possible to derive a ““Reynolds stress transport modelReynolds stress transport model”” by by

applying the time averaging operator as followsapplying the time averaging operator as follows

wherewhere

it is foundit is found

3535



•• This equation shows This equation shows the typical difficulties encountered when the typical difficulties encountered when

trying to trying to ““closeclose”” the turbulence equationsthe turbulence equations. In fact:. In fact:

�� the application of the timethe application of the time--averaging operator to the averaging operator to the NavierNavier--

Stokes equations makes the Reynolds stress tensor to Stokes equations makes the Reynolds stress tensor to

appear as a SECOND ORDER tensor of appear as a SECOND ORDER tensor of ““correlationcorrelation”” between between

two fluctuating velocity componentstwo fluctuating velocity components

�� the derivation of transport equations for the Reynolds stress the derivation of transport equations for the Reynolds stress

tensor makes tensor makes HIGHER ORDER correlation terms to appearHIGHER ORDER correlation terms to appear

•• The transport equation for turbulent kinetic energy can be obtaiThe transport equation for turbulent kinetic energy can be obtained ned

by taking the trace of the system of Reynolds stress transport by taking the trace of the system of Reynolds stress transport

equations; in factequations; in fact

3636



•• The k equation has the formThe k equation has the form

•• The Reynolds stress appearing in this equation has the formThe Reynolds stress appearing in this equation has the form

and the dissipation term has the formand the dissipation term has the form

and is evaluated by the relationshipand is evaluated by the relationship

3737



•• A A one equation model wasone equation model was proposed by proposed by PrandtlPrandtl in the formin the form

with with the additional closure equationthe additional closure equation

•• In general, oneIn general, one--equation models are incomplete, since the equation models are incomplete, since the turbulence length scale, turbulence length scale, llllllll , must be defined on a case by case basis; , must be defined on a case by case basis;

complete versions are anyway available which specify complete versions are anyway available which specify

independently this length (e.g., Baldwinindependently this length (e.g., Baldwin-- Barth, 1990).Barth, 1990).

•• In order to obtain complete models, In order to obtain complete models, an additional quantity must be an additional quantity must be

defineddefined also subjected to a transport equationalso subjected to a transport equation

3838



•• TwoTwo--equation modelsequation models are mostly based on the definition of this are mostly based on the definition of this

further quantity in the form of further quantity in the form of εεεεεεεε or or ω ω ω ω ω ω ω ω basing on the following basing on the following

relationships that relationships that ““closeclose”” the problem (other versions are available)the problem (other versions are available)

�� for for kk--ωωωωωωωω models it ismodels it is

in particular for the Wilcox (1998) model it isin particular for the Wilcox (1998) model it is

with appropriate values of the constants and, in particular:with appropriate values of the constants and, in particular:

3939



•• for for kk--εεεεεεεε models it ismodels it is

the dissipation equation can be derived exactly and has the the dissipation equation can be derived exactly and has the

classical formclassical form

The The standard standard kk--εεεεεεεε modelmodel adopts the definitionsadopts the definitions

4040



•• As presented, the above turbulence models are mostly suited for As presented, the above turbulence models are mostly suited for

dealing with turbulence conditions far from wallsdealing with turbulence conditions far from walls

•• When wall phenomena must be dealt withWhen wall phenomena must be dealt with two possible approaches two possible approaches

are available:are available:

�� use of use of ““wall functionswall functions””:: the logarithmic trend observed for the logarithmic trend observed for

velocity close to a flat surface is assumed to hold velocity close to a flat surface is assumed to hold

approximately near the specific considered wall, together approximately near the specific considered wall, together

with a corresponding temperature trend; with a corresponding temperature trend; in this case, the in this case, the

value of y+ in the first node close to the wall must be value of y+ in the first node close to the wall must be

conveniently large (e.g., y+ > 30conveniently large (e.g., y+ > 30););

�� use of low Reynolds number models:use of low Reynolds number models: these models are these models are

conceived to simulate the actual trend of turbulence close to conceived to simulate the actual trend of turbulence close to

the wall, by the adoption of the wall, by the adoption of damping functionsdamping functions; ; the value of the value of

y+ in the first node must be very small (typically y+

4141



•• On one hand, On one hand, the use of wall functions is computationally the use of wall functions is computationally

convenientconvenient, since refining the mesh close to the wall is expensive in , since refining the mesh close to the wall is expensive in

terms of resources (see the figure from terms of resources (see the figure from SharabiSharabi, 2008), 2008)

•• On the other hand, On the other hand, wall functions are not able to properly detect wall functions are not able to properly detect

some boundary layer phenomenasome boundary layer phenomena for which they were not for which they were not

conceived (e.g., buoyancy effects in heat transfer, etc.)conceived (e.g., buoyancy effects in heat transfer, etc.)

•• Nevertheless, even lowNevertheless, even low--Reynolds number models are not always Reynolds number models are not always

completely accuratecompletely accurate……

(a) Wall functions mesh (b) Low-Reynolds number mesh

4242


modelling of turbulent flowsmodelling of turbulent flowsDamping functions in lowDamping functions in low--Re modelsRe models

•• In In lowlow--Reynolds number modelsReynolds number models the definition of eddy viscosity is the definition of eddy viscosity is

changed from the classical formulationchanged from the classical formulation

to various forms including to various forms including damping functions, damping functions, ffµµµµµµµµ

that provide for that provide for the decrease of the eddy viscosity while the decrease of the eddy viscosity while

approaching the wallapproaching the wall

•• This allows This allows integration of the turbulence models through the integration of the turbulence models through the

boundary layer up to the wall itselfboundary layer up to the wall itself

•• Different assumptions lead to various formulations of the lowDifferent assumptions lead to various formulations of the low--Re Re

models and, generally, to different resultsmodels and, generally, to different results……

2

T C f kµ µν ε= 0 0f for yµ → →

4343


modelling of turbulent flowsmodelling of turbulent flowsLowLow--Re models vs. wall functionsRe models vs. wall functions

•• Providing an answer to Providing an answer to the questionthe question if the use of wall functions if the use of wall functions

should be preferred or notshould be preferred or not to models having a lowto models having a low--Re capabilityRe capability is is

not trivial, since:not trivial, since:

�� it heavily depends on the applicationit heavily depends on the application

�� it is strictly linked to the purpose of the analysisit is strictly linked to the purpose of the analysis

•• In this lecture I will propose In this lecture I will propose a case in which a case in which WFsWFs are not applicableare not applicable, ,

since they completely overlook phenomena related to buoyancysince they completely overlook phenomena related to buoyancy

•• In a lecture to come on condensation, In a lecture to come on condensation, I will show that the use of I will show that the use of

some minimum lowsome minimum low--Re number capabilities is useful to get relatively Re number capabilities is useful to get relatively

good agreement with experimental data though approximate good agreement with experimental data though approximate

method are also acceptablemethod are also acceptable; however, pending questions are: ; however, pending questions are:

�� could we afford describing a whole nuclear reactor could we afford describing a whole nuclear reactor

containment with such a strong refinement at the walls?containment with such a strong refinement at the walls?

�� couldncouldn’’t we instead accept a more approximate view of local t we instead accept a more approximate view of local

phenomena to get a reasonable overall picture?phenomena to get a reasonable overall picture?

4444


modelling of turbulent flowsmodelling of turbulent flowsAnisotropic RANS Anisotropic RANS -- 11

This choice is anyway heavy for the number of equations to be solved

A further possibility is to use an anisotropic RANS modelsin which the simple Boussinesq approximation is abandoned

�� The assumption of an isotropic value ofThe assumption of an isotropic value of ννννννννTT is not suitable for is not suitable for simulating details of flow in noncircular passagessimulating details of flow in noncircular passages

�� This is particularly true for This is particularly true for secondary flowssecondary flows in the direction in the direction

orthogonal to the main flow that would require the full orthogonal to the main flow that would require the full

Reynolds stress transport models to be predictedReynolds stress transport models to be predicted

RSM application from RSM application from SharabiSharabi (2008)(2008)

4545


modelling of turbulent flowsmodelling of turbulent flowsAnisotropic RANS Anisotropic RANS -- 22

In particular, it is possible to use In particular, it is possible to use algebraic expressionsalgebraic expressions of the kindof the kind

(see e.g., (see e.g., BagliettoBaglietto et al., 2006) which is limited to second order et al., 2006) which is limited to second order

terms in the strain and the rotational rates terms in the strain and the rotational rates SSijij and and ΩΩΩΩΩΩΩΩijij with respect with respect

to the original third order formulationto the original third order formulation

((BagliettoBaglietto et al., 2006)et al., 2006)

4646

TwoTwo--phase flow applicationsphase flow applicationsFew general considerationsFew general considerations

�� TwoTwo--phase flow introduces phase flow introduces additional complexityadditional complexity to the to the already complex problem of simulating turbulent flowalready complex problem of simulating turbulent flow

�� The presence of two phases and of The presence of two phases and of the related interfacesthe related interfacesrequires particular care in modellingrequires particular care in modelling

�� Ambitious goals of modelling twoAmbitious goals of modelling two--phase flow with CFD phase flow with CFD would be, for instance, to represent important phenomena would be, for instance, to represent important phenomena like CHF from first principleslike CHF from first principles

4747

TwoTwo--phase flow applicationsphase flow applicationsFew general considerations (contFew general considerations (cont’’d)d)

�� The work in the application of CFD techniques to twoThe work in the application of CFD techniques to two--phase flows phase flows

was developed for more than a decade, though nowadays it is stilwas developed for more than a decade, though nowadays it is still l

noted that the noted that the obtained models are not yet so mature as the ones obtained models are not yet so mature as the ones

for singlefor single--phase flows phase flows (foreword to (foreword to NuclNucl. Eng. Des., 240 (2010)). Eng. Des., 240 (2010))

�� The field is therefore one of active research, requiring The field is therefore one of active research, requiring huge huge

computational resources; computational resources; the brand name of Computational Multithe brand name of Computational Multi--

Fluid Dynamics (CMFD) was proposed for this field of research byFluid Dynamics (CMFD) was proposed for this field of research by

Prof. Prof. YadigarogluYadigaroglu (Int. J. (Int. J. MultiphMultiph. Flow, 23, 2003). Flow, 23, 2003)

�� In principle, DNS, LES and RANS techniques can be all usedIn principle, DNS, LES and RANS techniques can be all used for twofor two--

phase flowphase flow, though the scenario of their application is strongly , though the scenario of their application is strongly

changed with respect to singlechanged with respect to single--phasephase

�� In particular, in addition to the integral length scale and the In particular, in addition to the integral length scale and the

smallest turbulent scale, smallest turbulent scale, the scales of twothe scales of two--phase flow structuresphase flow structures

(e.g., bubbles) (e.g., bubbles) are called into playare called into play

4848


�� In the case of the In the case of the RANS approachRANS approach, , mass energy and momentum balance mass energy and momentum balance equationsequations are written in are written in 3D geometry3D geometry for each phase k (see e.g., for each phase k (see e.g., BestionBestionet al. 2005; et al. 2005; MimouniMimouni et al., 2008, et al., 2008, GalassiGalassi et al., 2009 for NEPTUNE)et al., 2009 for NEPTUNE)

�� These equations are accompanied by an extension to twoThese equations are accompanied by an extension to two--phase flow of phase flow of a a kk--εεεεεεεε modelmodel

where additional terms of where additional terms of turbulence productionturbulence production appear due to the appear due to the interaction between the phases. interaction between the phases.

An An interfacial area concentration transport equationinterfacial area concentration transport equation is also usedis also used

( ) kkkkkk wt

Γ=⋅∇+∂

∂ �ρα

ρα( ) ( )Tk k k k k k k k k k k k k k

ww w p M g

t

α ρα ρ α α ρ α τ τ

∂ + ∇ ⋅ = − ∇ + + + ∇ ⋅ + ∂

�

� ��

� � � � �

( )2 2 2

, , ,2 2 2

Tk k kk k k k k k k k k k k k k i k i i w k k k k

w w wph h w g w h q a q q q

t tα ρ α ρ α α ρ Γ α ∂ ∂

′′ ′′′+ + ∇ ⋅ + = + ⋅ + + + + − ∇ ⋅ + ∂ ∂

� � �

[ ],1

Production termsT

ik k k kk k i k k k K

i k j K j

k k kw P

t x x x

µρ α ρ ε

α σ

∂ ∂ ∂∂+ = + − +

∂ ∂ ∂ ∂

[ ], 1 11

C Production terms CT

ik k k k kk k i k k k

i k j j k

w Pt x x x k

ε ε ε

ε

ε ε µ ε ερ α ρ ε

α σ

∂ ∂ ∂∂+ = + − +

∂ ∂ ∂ ∂

4949


�� Needless to say, Needless to say, this model relies on the this model relies on the BoussinesqBoussinesqassumptionassumption; turbulent viscosity is moreover given simply by; turbulent viscosity is moreover given simply by

�� Its is quite clear that Its is quite clear that the success of such a model is strictly the success of such a model is strictly linked to its ingredients in terms of constitutive relationshipslinked to its ingredients in terms of constitutive relationshipsthat must be suitable for the particular considered flow regimethat must be suitable for the particular considered flow regime

�� In particular, for a bubbly flow the momentum transfer term, In particular, for a bubbly flow the momentum transfer term, MMk k , should account for , should account for mass transfermass transfer, the , the dragdrag and and liftlift forces, forces, the the addedadded mass termmass term and the and the turbulent dispersion of bubblesturbulent dispersion of bubbles

�� A major lack of RANS approaches is anyway in the fact that A major lack of RANS approaches is anyway in the fact that some twosome two--phase flow fields are naturally unstable: phase flow fields are naturally unstable: time time averaging is therefore suitable only to have a global averaging is therefore suitable only to have a global ““averagedaveraged””picturepicture of what happens, loosing instantaneous details (see of what happens, loosing instantaneous details (see e.g., the discussion in e.g., the discussion in YadigarogluYadigaroglu et al., 2008)et al., 2008)

k

kk

T

k

kC

ερµ µ

2

=

5050

�� By the way, unsteady calculations with RANS may show By the way, unsteady calculations with RANS may show

oscillations that may somehow match with experimental oscillations that may somehow match with experimental

observations (observations (ZborayZboray and De and De CahardCahard, 2005), 2005)

�� LES modelsLES models, of course, reintroduce the possibility to address , of course, reintroduce the possibility to address

varying flow fields like the fluctuations of bubble plumes; suchvarying flow fields like the fluctuations of bubble plumes; such

applications are interestingly discussed, among the others, by applications are interestingly discussed, among the others, by

YadigarogluYadigaroglu et al., (2008) and in works there referred to, and et al., (2008) and in works there referred to, and

by by NicenoNiceno et al., (2008)et al., (2008)

�� In such discussions, it can be noted that, in similarity with thIn such discussions, it can be noted that, in similarity with the e

case of RANS, case of RANS, LES models require accurate closure models for LES models require accurate closure models for

the different terms appearing in the equations in addition to the different terms appearing in the equations in addition to

adequate SGS modelsadequate SGS models


5151

�� LaheyLahey (2009) recently discussed the capabilities of (2009) recently discussed the capabilities of DNS DNS modelsmodels in representing twoin representing two--phase flowsphase flows

�� As in case of singleAs in case of single--phase flow, the attractiveness of this phase flow, the attractiveness of this technique lies in the fact that there is no need to technique lies in the fact that there is no need to introduce empirical models to obtain accurate introduce empirical models to obtain accurate predictions; the obvious drawback is the heavy predictions; the obvious drawback is the heavy computational loadcomputational load

�� In the case of twoIn the case of two--phase flows, phase flows, interface tracking interface tracking algorithmsalgorithms must be introduced; in the mentioned paper, must be introduced; in the mentioned paper, an algorithm based on the signed distance form the an algorithm based on the signed distance form the interface is used in the PHASTA codeinterface is used in the PHASTA code

�� Dam break problems, bubble interactions and plunging Dam break problems, bubble interactions and plunging jets are within the predictive capabilities, whenever jets are within the predictive capabilities, whenever appropriate computational resources are made availableappropriate computational resources are made available

CFDCFD--FigureFigure--2.ppt2.ppt


5252

Prediction of heat transfer deteriorationPrediction of heat transfer deteriorationAddressed experimental dataAddressed experimental data

� As in Sharabi et al. [2007], the considered experimental data are those by Pis’menny et al. [2006]:

– National Technological University of Ukraine

– turbulent heat transfer in vertical tubes for supercritical water

– operating pressure of 23.5 MPa

– inlet temperature and heating conditions involved in these analyses resulted in both dense and gas-like fluid to be present in the test section

– thin wall stainless steel tubes with inner diameters of 6.28 and 9.50 mm were adopted, with a 600 mm long heated section preceded by a 64 diameters long unheated region

– cromel-alumel thermocouples were adopted to measure the inlet and outlet fluid temperature, as well as the outer temperature of the tubes.

5353

Prediction of heat transfer deteriorationPrediction of heat transfer deteriorationPrevious resultsPrevious results

� Previous results obtained by Sharabi et al. [2007] with an in-house code

(AKN = Abe et al. [1994]; CH = Chien [1982]; JL = Jones and Launder [1972]; LB = Lam and Bremhorst, [1981]; LS = Launder and Sharma [1974]; YS = Yang and Shih [1993], WI=Wilcox [1994], SP=Speziale et al. [1990])

a) 6.28 mm ID, q”=390 kW/m

2, G= 590 kg/(m

2s),

Tinlet =300 °C, upward flow b) 6.28 mm ID, q”=390 kW/m

2, G= 590 kg/(m

2s),

Tinlet =300 °C, downward flow

5454

� It can be noted that:

– k-εεεε models predict in a qualitatively reasonable way the onset of heat transfer deterioration occurring in upward flow

– however, despite of quantitative differences between the results of the different k-εεεε models, they all tend to predict a larger wall temperature increase than observed

– on the other hand, the Wilcox [1994] k-ωωωω model (WI) and the Speziale et al. [1990] k-ττττ model (SP) were seen to predict no deterioration or a very delayed one

– in the case of upward flow, all the models provided similar results, characterised by the absence of any deterioration phenomenon, in qualitative agreement with experimental observations

Prediction of heat transfer deteriorationPrediction of heat transfer deteriorationPrevious results (contPrevious results (cont’’d)d)

5555

Velocity distribution predicted by the YS model

(upward flow, G=509 kg/(m2s), q=390 kW/m2,

tin=300 °C)

Velocity distribution predicted by the

WI model (upward flow, with G=509

kg/(m2s), q=390 kW/m2, tin=300 °C)

(Longer pipe)

Buoyancy forces accelerate the flow at the wall and lead to an “m-shaped velocity

profile”

Reasons for HeatTransfer

Deterioration


5656

Turbulent kinetic energy distribution predicted

by the YS model (upward flow, G=509 kg/(m2s),

q=390 kW/m2, tin=300 °C)

Turbulent kinetic energy distribution

predicted by the WI model (upward

flow, G=509 kg/(m2s), q=390 kW/m2,

tin=300 °C)

(Longer pipe)

In the transition to the “m-shaped profile” velocity gradients are suppressed and turbulence production decreases


5757

� With the STAR-CCM+ code, the following modelling choices were made:– The adopted 2D axi-symmetric mesh included

� 20 radial nodes in a 0.54 mm thick prismatic layer region close to the wall

� 26 uniform nodes in the remaining core region, having a radius of 2.6 mm

� The stretching factor adopted in the prismatic layer was 1.2

� “Trimmed” meshes were selected for the core region

– Though slightly coarser than in the in-house code calculations, the grid was found to be suitable to provide enough accurate results with a reasonable computational effort

– Later, the results obtained by this grid have been compared to those obtained by a finer one (68 radial and 500 axial nodes) showing little differences

– Default code options were adopted in relation to advection schemes (2nd order)

– The steady-state iteration algorithm of the code was adopted, starting with coupled flow and energy iterations and then shifting to thesegregated equation approach

– In all the code runs, it was checked that the requirement y+ < 1 was respected with due margin

Prediction of heat transfer deteriorationPrediction of heat transfer deteriorationSTARSTAR--CCM+ ResultsCCM+ Results

5858

� Concerning water properties at 23.5 MPa, the code allows assigning the dependence of density and specific heat on temperature in polynomial form

� Thermal conductivity and dynamic viscosity can be instead assigned adopting user defined field functions.

� Suitable local cubic spline polynomials were then used for these properties, whose coefficients were generated on the basis of tables obtained by the NIST package

0

200

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Data

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Interval Boundaries

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)]

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Prediction of heat transfer deteriorationPrediction of heat transfer deteriorationSTARSTAR--CCM+ Results (contCCM+ Results (cont’’d)d)

5959

The analysis reported herein was limited to four k-εεεε models:

� the Two-Layer All y+ Wall Treatment (referred to in the following as “all y+”), suggested for simulating with a reasonable accuracy different kinds of flows;

� the standard Low-Reynolds Number K-Epsilon Model (referred to in the following as “low-Re”) suggested by code guidelines for natural convection problems and referred to a model published by Lien etal. [1996];

� the AKN model, already used with the in-house code [Abe et al., 1994];

� the V2F model that, besides the k and εεεε equations, solves two additional transport and algebraic equations; this model is suggested to capture more accurately near wall phenomena [Durbin, 1991; Durbin, 1996; Lien et al., 1998].


6060

300

400

500

600

700

800

900

0 20 40 60 80 100

x / D

Wall

Tem

per

atu

re [

°C]

Low-Re

AKN

V2F

All y+

Low-Re (finer mesh)

Experiment

a) 6.28 mm ID, q”=390 kW/m

2, G= 590 kg/(m

2s),

Tinlet =300 °C, upward flow


6161

300

400

500

600

700

800

900

0 20 40 60 80 100

x / D

Wa

ll T

emp

eratu

re [

°C]

Low-Re

AKN

V2F

All y+

Experiment

a) 6.28 mm ID, q”=390 kW/m

2, G= 590 kg/(m

2s),

Tinlet =300 °C, downward flow


6262

It can be noted that:

� the Two-Layer All y+ Wall Treatment was unable to detect the start of deterioration phenomena in upward flow

� all the other k-εεεε models showed a behaviour similar to the one already observed in the previous study:– they are able to detect the onset of deterioration– they tend to overestimate the effect of deterioration on wall temperature prediction

� all the models have no difficulty to predict the behaviourobserved in downward flow, in which no deterioration was detected

The reasons of this behaviour were found to be the same as observed in the previous study (see below)


6363

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elo

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mp

on

en

t [m

/s] Pipe Inlet

0

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64

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All y+ Model, Upward Flow

x/D

Figure 1: Radial distribution of the axial velocity component in the upward flow case


6464

0.000

0.001

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rbu

len

t K

inet

ic E

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[J/k

g]

Pipe Inlet

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rbu

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inet

ic E

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g]

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ic E

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g]

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x/D

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Radius [m]

Tu

rbu

len

t K

inet

ic E

ner

gy

[J/k

g]

Pipe Inlet

0

16

32

48

64

80

88

All y+ Model, Upward Flow

x/D

Figure 1: Radial distribution of turbulent kinetic energy in the upward flow case


6565

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Figure 1: Radial distribution of the axial velocity component in the downward flow case


6666

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x/D

Figure 1: Radial distribution of turbulent kinetic energy in the downward flow case


6767

CFD and CMFD are very powerful tools, whose capabilities are conditioned to our understanding of phenomena and to computer power

The smaller is the degree of empiricism we wish to introduce in the models, the greatest is the computer power needed

It is a very fascinating world in which smart ideas are needed to discover newer and newer possibilities

In summaryIn summary……

6868

ThankThankThankThankThankThankThankThank youyouyouyouyouyouyouyou forforforforforforforfor youryouryouryouryouryouryouryour attentionattentionattentionattentionattentionattentionattentionattention,,,,,,,,

Walter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter Ambrosini

6969

Sources and suggested readingsSources and suggested readings

•• N.E. N.E. TodreasTodreas, M. S. , M. S. KazimiKazimi ““Nuclear Systems INuclear Systems I””, Taylor & Francis, 1990., Taylor & Francis, 1990.

•• D.J. D.J. TrittonTritton ““Physical Fluid DynamicsPhysical Fluid Dynamics””, Oxford Science Publications, 2, Oxford Science Publications, 2ndnd Edition, 1997.Edition, 1997.

•• H.K. H.K. VeerstegVeersteg and W. and W. MalalasekeraMalalasekera ““An introduction to computational fluid dynamicsAn introduction to computational fluid dynamics””, Pearson, Prentice Hall, 1995., Pearson, Prentice Hall, 1995.

•• D.C. Wilcox D.C. Wilcox ““Turbulence Turbulence ModelingModeling for CFDfor CFD””, 2nd Edition, DCW Industries, 1998., 2nd Edition, DCW Industries, 1998.

•• E. E. BagliettoBaglietto, H. , H. NinokataNinokata, , TakeharuTakeharu MisawaMisawa, , CFD and DNS methodologies development for fuel bundle simulationCFD and DNS methodologies development for fuel bundle simulations, Nuclear s, Nuclear

Engineering and Design 236 (2006) 1503Engineering and Design 236 (2006) 1503––15101510

•• Maria Cristina Maria Cristina GalassiGalassi, Pierre , Pierre CosteCoste, Christophe Morel and Fabio , Christophe Morel and Fabio MorettiMoretti, Two, Two--Phase Flow Simulations for PTS Phase Flow Simulations for PTS

Investigation by Means of Neptune CFD Code, Investigation by Means of Neptune CFD Code, HindawiHindawi Publishing Corporation, Science and Technology of Nuclear Publishing Corporation, Science and Technology of Nuclear

Installations, Volume 2009, Article ID 950536, 12 pages, doi:10.Installations, Volume 2009, Article ID 950536, 12 pages, doi:10.1155/2009/9505361155/2009/950536

•• D. D. BestionBestion and A. and A. GuelfiGuelfi, Status and Perspective of Two, Status and Perspective of Two--Phase Flow Phase Flow ModellingModelling in the Neptune in the Neptune MultiscaleMultiscale ThiermalThiermal--

Hydraulic Platform for Nuclear Reactor Simulation, NUCLEAR ENGINHydraulic Platform for Nuclear Reactor Simulation, NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.37 NO.6 EERING AND TECHNOLOGY, VOL.37 NO.6

DECEMBER 2005DECEMBER 2005

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