lappeenranta wa 1 cfd

7/30/2019 Lappeenranta WA 1 CFD

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Basic Concepts about CFD ModelsBasic Concepts about CFD Models

Walter AmbrosiniWalter Ambrosini

Associate Professor inAssociate Professor inNuclearNuclearPlantsPlants

at theat theUniversityUniversityofofPisaPisa

Lappeenranta University of TechnologyLappeenranta University of Technology

Summer School in Heat and Mass TransferSummer School in Heat and Mass TransferAugust 18August 1820, 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 AveragedReynolds Averaged NavierNavier--Stokes equations (RANS)Stokes equations (RANS)

TwoTwo--phase flow applicationsphase flow applications

Prediction of heat transfer deteriorationPrediction of heat transfer deterioration


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

The transition from laminar flow to turbulence isThe transition from laminar flow to turbulence is an example ofan example offlow instabilityflow instability::

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

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

this was shown bythis was shown by Osborne ReynoldsOsborne Reynolds in a classicalin a classical

experimentexperiment


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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



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In order to study stability of a nonlinear system by analyticalIn order to study stability of a nonlinear system by analyticalmeans the methodology ofmeans the methodology of linear stability analysislinear stability analysis is oftenis often

adoptedadopted

This has the objective to determineThis has the objective to determine the stability conditionsthe 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 TRANSIENTEXAMPLES OF TRANSIENT

ANALYSESANALYSES

CavityCavityRB ConvectionRB Convection

Buoyant JetBuoyant Jet


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Turbulence introduces a large degree ofTurbulence introduces a large degree of sensitivity to initialsensitivity to initialconditions (SIC)conditions (SIC) that is typical ofthat is typical of deterministic chaosdeterministic chaos

By this, it is meant thatBy this, it is meant that turbulent motion is notturbulent motion is not randomrandom,,

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

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

This characteristic is shared with many differentThis characteristic is shared with many different chaoticchaotic

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


dRe

d= Gr

1

2-

L

Df'(Re) Re |Re|

d1

d= Re 1 -

2 Fo 1 +4

sin

d1

d = - Re 1 - 2 Fo 1 +4

cos Heating

Cooling


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Owing to the fluctuating nature of the turbulent flow field, itOwing to the fluctuating nature of the turbulent flow field, it isiscustomary (after Reynolds)customary (after Reynolds) to introduce an appropriate timeto 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 writeThe attempt is quite evidently to write equations in terms ofequations 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, butThis attempt is successful, but fluctuations cannot be forgottenfluctuations cannot be forgotten

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


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In particular,In particular,

the following quantities have overwhelmingthe following quantities have overwhelming

importanceimportance

Turbulence intensity is strictly related to the turbulence kinetTurbulence intensity is strictly related to the turbulence kineticic

energyenergy

This is one of the most important quantities adopted in presentThis 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



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The general balance equations in local and instantaneousThe general balance equations in local and instantaneousformulation are then averagedformulation are then averaged making use of the abovemaking use of the above

described averaging operatordescribed averaging operator

After simplifications (described in lecture notes), an averagedAfter 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 equationss

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

This term, having a clearThis term, having a clear advectiveadvective nature, points out thatnature, 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 additionalsort of additional mixingmixing due to turbulencedue to turbulence



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In analogy with the molecular motion, the basic idea is thereforIn analogy with the molecular motion, the basic idea is thereforeeto interpret such term as anto interpret such term as an additional diffusion due toadditional diffusion due to

turbulenceturbulence

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

that calls for a proper modellingthat calls for a proper modelling



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TheThe Reynolds stress tensorReynolds stress tensor appears in momentum equationsappears in momentum equations

The Reynolds stresses account for the additional momentumThe 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 FlowMomentum Transfer in Turbulent Flow -- 11


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It is then customary to adopt theIt is then customary to adopt the BoussinesqBoussinesq approximationapproximationbased on a definition ofbased 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 TT is no more a property of the fluid, but alsois no more a property of the fluid, but also

depends on flow.depends on flow.

Of course,Of course, thethe BoussinesqBoussinesq approximation shifts the toughnessapproximation 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 beBy the way, many different kinds of turbulence can be

envisaged, ranging from ideally homogeneous and isotropic toenvisaged, 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 toEddy viscosity models have therefore the very tough job to

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

approximationapproximation



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It is rather instructive and useful to considerIt is rather instructive and useful to consider the distribution ofthe distribution ofvelocity 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

AA universal logarithmic velocity profileuniversal logarithmic velocity profile is found both on theis 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 beThe effect of turbulence in the transport of momentum can beclearly seen in comparing the distributions of velocity in theclearly 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 thee

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

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

turbulenceturbulence



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TheThe averaged total energy equationaveraged total energy equation and theand the steady thermalsteady thermalenergy 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 similarar

as in the case of momentumas in the case of momentum

wherewhere TT is theis the turbulent thermal diffusivityturbulent thermal diffusivity

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


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The picture of the turbulent transfer phenomenon is thereforeThe picture of the turbulent transfer phenomenon is thereforethe 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



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A simple but effective way to establish this relationship is toA simple but effective way to establish this relationship is todefine a constantdefine a constant turbulentturbulent PrandtlPrandtl numbernumber,, in analogy within analogy with

the molecular one assuming that, as a consequence of thethe 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 casein this case is that the same coherentis that the same coherent

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

transfertransfer

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

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

different approaches should be useddifferent approaches should be used



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In turbulent flow anIn turbulent flow an energy cascadeenergy cascade occurs representing theoccurs representing thetransfer of turbulence kinetic energy from larger to smallertransfer of turbulence kinetic energy from larger to smaller

eddieseddies

Basic concepts about computationalBasic concepts about computational

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

As such, turbulence can beAs such, turbulence can be

considered asconsidered as a phenomenona phenomenon

characterised by a wide range ofcharacterised by a wide range of

lengthslengths at which interestingat which interesting

phenomena do occur:phenomena do occur:

fromfrom the integral lengththe integral length

scalescale,, llllllll, at which energy is, at which energy isextracted from the mean flowextracted from the mean flow

toto thethe KolmogorovKolmogorov lengthlength

scalescale,, , at which turbulence, at which turbulence

kinetic energy is finallykinetic energy is finally

dissipated into heatdissipated into heat


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It must be noted that theIt must be noted that the KolmogorovKolmogorov length scale,length scale, ,,

is smallis smallbut still large with respect to the molecularbut still large with respect to the molecular mean free pathmean free path::

so, turbulence can still be studiedso, 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 becharacterising large eddies can bedefined as the average length over which a fluctuatingdefined as the average length over which a fluctuating

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

negligiblenegligible

On both dimensional and experimental basis, it can be shownOn both dimensional and experimental basis, it can be shownthatthat

andand

withwith ; 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 shouldan adequate representation of turbulence should take intotake into

account the phenomena of production and dissipation ofaccount 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 differentsimulating the transient evolution of vortices of different

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

scalescale (DNS, LES, DES)(DNS, LES, DES)

simulating turbulence on the basis of the above describedsimulating turbulence on the basis of the above describedstatistical approachstatistical approach, introducing appropriate production, introducing appropriate production

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

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




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

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

routes,routes, trying to simulate with the highest possible space andtrying to simulate with the highest possible space andtime 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 theThe assumption behind this technique is that the NavierNavier--StokesStokes

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

behaviour with no need of additional constitutive laws; forbehaviour with no need of additional constitutive laws; forincompressible flow it is:incompressible flow it is:

The web is full of fascinating pictures and movies about DNSThe 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 isThe application of this technique is very demanding in terms ofvery demanding in terms of

computational resourcescomputational resources: representing flows of technical: representing flows of technicalinterest is very challenging and requires massive parallelinterest is very challenging and requires massive parallel

computingcomputing

However the technique is very promising and it isHowever the technique is very promising and it is sometimessometimesused 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 whichIn 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 isDNS is

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

turbulence model can be validatedturbulence model can be validatedCFDCFD--FigureFigure--1.ppt1.ppt


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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 simulatingLES is aimed at simulating

only larger eddies, while the smaller scales are treated byonly larger eddies, while the smaller scales are treated bysubgridsubgrid--scale (SGS) modelsscale (SGS) models

In other words, there areIn 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 andAs 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 theA key point in LES is introducing a spatial filtering for the

smaller scalessmaller scales


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


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Once the resolvable scales are defined, the averaged NOnce the resolvable scales are defined, the averaged N--S equationsS equationsare written in averaged formare written in averaged form


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

or alsoor also

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

different definitions)different definitions)

it can be finally obtainedit can be finally obtained


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

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

equationsequations

where Cwhere CSS is theis the SmagorinskySmagorinsky coefficient representing a parameter tocoefficient representing a parameter tobe adjusted for the particular problem to be dealt with; valuesbe adjusted for the particular problem to be dealt with; values in thein the

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

LESLES is presently promising as a design tool, but still heavy from this presently promising as a design tool, but still heavy from thee

computational point of viewcomputational point of view


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

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

momentum equations in which turbulence is represented bymomentum equations in which turbulence is represented by thetheReynolds stressReynolds stress

TheThe 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 constanttant

turbulentturbulent PrandtlPrandtl number, also the eddy thermal diffusivity is related tonumber, 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 x

= = +


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

on the basis of the number of the additional partial differentiaon the basis of the number of the additional partial differentiallequations 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 betweenone between complete and incomplete modelscomplete and incomplete models::

completenesscompleteness of the model is related to its capability toof 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 boundaryin 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 thehemixing length theory ofmixing length theory of PrandtlPrandtl (1925)(1925)

wherewhere llllllllmixmix is the mixing length; the model is similar to the one foris the mixing length; the model is similar to the one for

molecular viscositymolecular viscosity in which kinematic viscosity is a interpreted asin 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 freeree

path)path)

In the presence of a wall, it is assumedIn the presence of a wall, it is assumed where the constantwhere the constantmust 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, butbut

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

equations to be preferableequations to be preferable


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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 ait is possible to derive a Reynolds stress transport modelReynolds stress transport model byby

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

wherewhere

it is foundit is found


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This equation showsThis equation shows the typical difficulties encountered whenthe typical difficulties encountered when

trying totrying to closeclose the turbulence equationsthe turbulence equations. In fact:. In fact: the application of the timethe application of the time--averaging operator to theaveraging operator to the NavierNavier--

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

appear as a SECOND ORDER tensor ofappear as a SECOND ORDER tensor of correlationcorrelation betweenbetween

two fluctuating velocity componentstwo fluctuating velocity components

the derivation of transport equations for the Reynolds stressthe derivation of transport equations for the Reynolds stresstensor makestensor 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 obtainedned

by taking the trace of the system of Reynolds stress transportby taking the trace of the system of Reynolds stress transportequations; in factequations; in fact


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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


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AA one equation model wasone equation model was proposed byproposed by PrandtlPrandtl in the formin the form

withwith the additional closure equationthe additional closure equation

In general, oneIn general, one--equation models are incomplete, since theequation 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 specifycomplete 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 bean additional quantity must be

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


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

further quantity in the form offurther quantity in the form of oror basing on the followingbasing on the followingrelationships thatrelationships that closeclose the problem (other versions are available)the problem (other versions are available)

forfor kk-- 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:


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forfor kk-- models it ismodels it is

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

classical formclassical form

TheThe standardstandard kk-- modelmodel adopts the definitionsadopts the definitions


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As presented, the above turbulence models are mostly suited forAs 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 approachestwo possible approaches

are available:are available:

use ofuse of wall functionswall functions:: the logarithmic trend observed forthe logarithmic trend observed for

velocity close to a flat surface is assumed to holdvelocity close to a flat surface is assumed to holdapproximately near the specific considered wall, togetherapproximately near the specific considered wall, together

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

value of y+ in the first node close to the wall must bevalue 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 arethese models areconceived to simulate the actual trend of turbulence close toconceived to simulate the actual trend of turbulence close to

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

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


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On one hand,On one hand, the use of wall functions is computationallythe 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 interms of resources (see the figure fromterms of resources (see the figure from SharabiSharabi, 2008), 2008)

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

some boundary layer phenomenasome boundary layer phenomena for which they were notfor 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 alwaysReynolds number models are not always

completely accuratecompletely accurate

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


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

InIn lowlow--Reynolds number modelsReynolds number models the definition of eddy viscosity isthe definition of eddy viscosity is

changed from the classical formulationchanged from the classical formulation

to various forms includingto various forms including damping functions,damping functions, ff

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

approaching the wallapproaching the wall

This allowsThis allows integration of the turbulence models through theintegration 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--ReRe

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

2T C f k = 0 0f for y


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

Providing an answer toProviding an answer to the questionthe question if the use of wall functionsif 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 isisnot 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 proposeIn this lecture I will propose a case in whicha 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 ofI will show that the use of

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

good agreement with experimental data though approximategood agreement with experimental data though approximatemethod are also acceptablemethod are also acceptable; however, pending questions are:; however, pending questions are:

could we afford describing a whole nuclear reactorcould 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?

couldncouldnt we instead accept a more approximate view of localt we instead accept a more approximate view of local

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


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

This choice is anyway heavyfor the number of equationsto be solved

A further possibility is to usean anisotropic RANS modelsin which the simpleBoussinesq approximation isabandoned

The assumption of an isotropic value ofThe assumption of an isotropic value of TT is not suitable foris not suitable forsimulating details of flow in noncircular passagessimulating details of flow in noncircular passages

This is particularly true forThis is particularly true for secondary flowssecondary flows in the directionin the direction

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

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

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


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

In particular, it is possible to useIn 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 orderet al., 2006) which is limited to second order

terms in the strain and the rotational ratesterms in the strain and the rotational rates SSijij andand ijij with respectwith respect

to the original third order formulationto the original third order formulation

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


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

TwoTwo--phase flow introducesphase flow introduces additional complexityadditional complexity to theto the

already complex problem of simulating turbulent flowalready complex problem of simulating turbulent flow

The presence of two phases and ofThe 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 CFDphase flow with CFDwould be, for instance, to represent important phenomenawould be, for instance, to represent important phenomenalike CHF from first principleslike CHF from first principles


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

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

was developed for more than a decade, though nowadays it is stilwas developed for more than a decade, though nowadays it is stil llnoted that thenoted that the obtained models are not yet so mature as the onesobtained models are not yet so mature as the ones

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

The field is therefore one of active research, requiringThe field is therefore one of active research, requiring hugehuge

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 stronglychanged with respect to singlechanged with respect to single--phasephase

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


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In the case of theIn the case of the RANS approachRANS approach,, mass energy and momentum balancemass energy and momentum balanceequationsequations are written inare written in 3D geometry3D geometry for each phase k (see e.g.,for each phase k (see e.g., BestionBestion

et 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 ofphase flow ofaakk-- modelmodel

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

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

( ) kkkkkk w

t=+

( ) ( )Tk k k k k k k k k k k k k k

ww w p M g

t

+ = + + + +

( )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 t

+ + + = + + + + + +

[ ],1

Production terms

Tik k k k

k k i k k k K

i k j K j

k k kw P

t x x x

+ = + +

[ ], 1 11

C Production terms C

Tik k k k k

k k i k k k

i k j j k

w Pt x x x k

+ = + +


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Needless to say,Needless to say, this model relies on thethis model relies on the BoussinesqBoussinesq

assumptionassumption; turbulent viscosity is moreover given simply by; turbulent viscosity is moreover given simply by

Its is quite clear thatIts is quite clear that the success of such a model is strictlythe success of such a model is strictlylinked to its ingredients in terms of constitutive relationshipslinked to its ingredients in terms of constitutive relationships

that 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,

MMkk, should account for, should account for mass transfermass transfer, the, the dragdrag andand liftlift forces,forces,thethe addedadded mass termmass term and theand the turbulent dispersion of bubblesturbulent dispersion of bubbles

A major lack of RANS approaches is anyway in the fact thatA major lack of RANS approaches is anyway in the fact thatsome twosome two--phase flow fields are naturally unstable:phase flow fields are naturally unstable: timetimeaveraging is therefore suitable only to have a globalaveraging is therefore suitable only to have a globalaveragedaveragedpicturepicture of what happens, loosing instantaneous details (seeof what happens, loosing instantaneous details (seee.g., the discussion ine.g., the discussion inYadigarogluYadigaroglu et al., 2008)et al., 2008)

k

kk

T

k

kC

2

=


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

oscillations that may somehow match with experimentaloscillations that may somehow match with experimentalobservations (observations (ZborayZboray and Deand 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, byapplications are interestingly discussed, among the others, by

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

byby 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 theecase of RANS,case of RANS, LES models require accurate closure models forLES models require accurate closure models for

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

adequate SGS modelsadequate SGS models



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LaheyLahey (2009) recently discussed the capabilities of(2009) recently discussed the capabilities of DNSDNS

modelsmodels in representing twoin representing two--phase flowsphase flows As in case of singleAs in case of single--phase flow, the attractiveness of thisphase flow, the attractiveness of this

technique lies in the fact that there is no need totechnique lies in the fact that there is no need tointroduce empirical models to obtain accurateintroduce empirical models to obtain accuratepredictions; the obvious drawback is the heavypredictions; the obvious drawback is the heavy

computational loadcomputational load

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

Dam break problems, bubble interactions and plungingDam break problems, bubble interactions and plungingjets are within the predictive capabilities, wheneverjets are within the predictive capabilities, wheneverappropriate computational resources are made availableappropriate computational resources are made available

CFDCFD--FigureFigure--2.ppt2.ppt



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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 Pismenny et al. [2006]:

National Technological University of Ukraine

turbulent heat transfer in vertical tubes for supercriticalwater

operating pressure of23.5 MPa inlet temperature and heating conditions involved in these

analyses resulted in both dense and gas-like fluid to bepresent in the test section

thin wall stainless steel tubes with inner diameters of 6.28and 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 outertemperature of the tubes.


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Prediction of heat transfer deteriorationPrediction of heat transfer deteriorationPrevious resultsPrevious results

Previous results obtained by Sharabi et al. [2007] with anin-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 flowb) 6.28 mm ID, q=390 kW/m

2, G= 590 kg/(m

2s),

Tinlet =300 C, downward flow


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It can be noted that:

k- models predict in a qualitatively reasonable way the onsetof heat transfer deterioration occurring in upward flow

however, despite of quantitative differences between theresults of the different k- models, they all tend to predict a

larger wall temperature increase than observed on the other hand, the Wilcox [1994] k- model (WI) and the

Speziale et al. [1990] k- model (SP) were seen to predict nodeterioration or a very delayed one

in the case of upward flow, all the models provided similar

results, characterised by the absence of any deteriorationphenomenon, in qualitative agreement with experimentalobservations

Prediction of heat transfer deteriorationPrediction of heat transfer deteriorationPrevious results (contPrevious results (contd)d)


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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

WImodel (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 leadto an m-shaped velocityprofile

Reasons for Heat

TransferDeterioration



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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 WImodel (upward

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

tin=300 C)

(Longer pipe)

In the transition to the m-shapedprofile velocity gradients are

suppressed and turbulenceproduction decreases



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With the STAR-CCM+ code, the following modelling choices weremade:

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 2.6mm

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 gridwas found to be suitable to provide enough accurate results with areasonable computational effort

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

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 wasrespected with due margin

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


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Concerning water properties at 23.5 MPa, the code allows assigningthe 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 theseproperties, whose coefficients were generated on the basis of tablesobtained by the NIST package

0

200

400

600

800

1000

1200

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Temperature [K]

Density[kg/m

3]

Data

Splines

Interval Boundaries

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Temperature [K]

C

p[J/(kgK)]

Data

Splines

Interval Boundaries

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Temperature [K]

ThermalC

onductivity[W/(mK)]

Data

Splines

Interval Boundaries

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

1.6E-03

1.8E-03

2.0E-03

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Temperature [K]

DynamicViscosity[kg/(ms)]

Data

Splines

Interval Boundaries

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

640 645 650 655 660 665 670 675 680

Temperature [K]

C

p[J/(kgK)]

Data

Splines

Interval Boundaries

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

640 650 660 670 680 690 700

Temperature [K]

ThermalC

onductivity[W/(mK)]

Data

Splines

Interval Boundaries

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


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

the Two-Layer All y+ Wall Treatment (referred to in the following asall y+), suggested for simulating with a reasonable accuracydifferent kinds of flows;

the standard Low-Reynolds Number K-Epsilon Model (referred to in

the following as low-Re) suggested by code guidelines for naturalconvection 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 equations, solves twoadditional transport and algebraic equations; this model issuggested to capture more accurately near wall phenomena[Durbin, 1991; Durbin, 1996; Lien et al., 1998].



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300

400

500

600

700

800

900

0 20 40 60 80 100

x / D

WallTemperature[C]

Low-ReAKN

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



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300

400

500

600

700

800

900

0 20 40 60 80 100

x / D

WallTem

perature[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



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It can be noted that:

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

all the other k- models showed a behaviour similar to theone 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 asobserved in the previous study (see below)



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0

0.2

0.4

0.6

0.8

1

1.2

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

X-VelocityComponent[m/s] Pipe Inlet

0

16

32

48

64

80

88

Low-Re Model, Upward Flow

x/D

0

0.2

0.4

0.6

0.8

1

1.2

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]


0

16

32

48

64

80

88

AKN Model, Upwar d Flow

x/D

0

0.2

0.4

0.6

0.8

1

1.2

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

X-VelocityCompo

nent[m/s] Pipe Inlet

0

16

3248

64

80

88

V2F Model, Upward Flow

x/D

0

0.2

0.4

0.6

0.8

1

1.2

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

X-VelocityCompo


0

16

3248

64

80

88

All y+ Model, Upward Flow

x/D

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



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0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

TurbulentKineticEnergy[J/kg]

Pipe Inlet

0

16

32

48

64

80

88

Low-Re Model, Upward Flow

x/D

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]


Pipe Inlet

0

16

32

48

64

80

88

AKN Model, Upward Flow

x/D

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

TurbulentKineticEn

ergy[J/kg] Pipe Inlet

0

16

3248

64

80

88

V2F Model, Upward Flow

x/D

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

TurbulentKineticEn

ergy[J/kg] Pipe Inlet

0

16

3248

64

80

88

All y+ Model, Upward Flow

x/D

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


P di ti f h t t f d t i ti


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]


0

16

32

48

64

80

88

Low-Re Model, Downward Flow

x/D

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]


0

16

32

48

64

80

88

AKN Model, Downward Flow

x/D

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

X-VelocityCompo


0

16

3248

64

80

88

V2F Model, Downward Flow

x/D

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

X-VelocityCompo


0

16

3248

64

80

88

All y+ Model, Downw ard Flow

x/D

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


P di ti f h t t f d t i ti


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6666

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]


Pipe Inlet0

16

32

48

64

80

88

Low-Re Model, Downward Flow

x/D

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]


Pipe Inlet0

16

32

48

64

80

88

AKN Model, Downward Flow

x/D

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

TurbulentKineticE

nergy[J/kg] Pipe Inlet

0

16

32

48

64

80

88

V2F Model, Downward Flow

x/D

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Radius [m]

TurbulentKineticE

nergy[J/kg] Pipe Inlet

0

16

32

48

64

80

88

All y+ Model, Downward Flow

x/D

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


In summary


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CFD and CMFD are very powerful tools, whose

capabilities are conditioned to our understanding ofphenomena and to computer power

The smaller is the degree of empiricism we wish tointroduce in the models, the greatest is the computerpower needed

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

In summaryIn summary


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

Walter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter AmbrosiniWalter Ambrosini


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