development and assessment of computational fluid … fanale_definitiva.pdfdevelopment and...

POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science in Nuclear Engineering

Development and assessment of Computational

Fluid Dynamics models for the study of natural

circulation dynamics

Supervisor: Prof. Lelio Luzzi

Assistant supervisor: Ing. Alessandro Pini

Author: Francesco Fanale Matr. 818262

Academic year

2014-2015

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CONTENTS

CONTENTS ................................................................................................................. III

ACKNOWLEDGEMENTS ......................................................................................... VII

ABSTRACT ...................................................................................................................IX

SOMMARIO ..................................................................................................................XI

ESTRATTO IN ITALIANO ....................................................................................... XIII

INTRODUZIONE ........................................................................................................ XIII

STUDIO NUMERICO DELLA CAVITÀ 2D RISCALDATA DALL’ESTERNO, NEL CASO DI

FLUIDI CON GENERAZIONE INTERNA DI ENERGIA, A DIVERSI NUMERI DI RAYLEIGH .. XV

STUDIO DEL COMPORTAMENTO DINAMICO DEI CIRCUITI A CIRCOLAZIONE NATURALE

CONVENZIONALE ....................................................................................................... XX

STUDIO DEL COMPORTAMENTO DINAMICO DEI CIRCUITI A CIRCOLAZIONE NATURALE

IN PRESENZA DI UNA SORGENTE INTERNA DI CALORE: IL CIRCUITO DI DYNASTY . XXIV

CONCLUSIONI ........................................................................................................ XXIX

NOMENCLATURE .............................................................................................. XXXIII

LIST OF SYMBOLS ................................................................................................. XXXIII

SPECIAL CHARACTERS .......................................................................................... XXXIV

SUBSCRIPTS-SUPERSCRIPTS ................................................................................. XXXIV

ACRONYMS ............................................................................................................ XXXV

LIST OF FIGURES .............................................................................................. XXXIX

LIST OF TABLES ................................................................................................. XLVII

INTRODUCTION .......................................................................................................... 3

REFERENCES .............................................................................................................. 6

CHAPTER 1: BRIEF OVERVIEW OF COMPUTATIONAL FLUID DYNAMICS .....11

1.1 INTRODUCTION ....................................................................................................11

1.2 NAVIER-STOKES AND ENERGY EQUATIONS ...........................................................11

1.3 TURBULENCE: THEORY AND MODELS ...................................................................13

1.3.1 Characteristics of turbulence .............................................................................. 13

1.3.2 Turbulence models ............................................................................................... 15

1.3.3 RANS models ....................................................................................................... 15

1.3.4 Equations for kinetic energy ............................................................................... 17

1.3.5 Standard 𝑘 − 휀 Model .......................................................................................... 18

1.3.6 Wilcox 𝑘 − 𝜔 Model .............................................................................................. 20

1.3.7 Menter Shear Stress Transport 𝑘 − 𝜔 ................................................................. 21

1.3.8 Near-wall treatment ............................................................................................ 22

1.4 FINITE VOLUME METHOD .....................................................................................23

1.5 THE OPENFOAM® CODE ......................................................................................24

CONTENTS

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1.5.1 OpenFOAM® structure ....................................................................................... 24

1.5.2 OpenFOAM® solvers ........................................................................................... 25

REFERENCES ............................................................................................................. 26

CHAPTER 2: NATURAL CIRCULATION OF FLUIDS CHARACTERIZED BY AN

INTERNAL ENERGY SOURCE IN A SQUARE CAVITY AT DIFFERENT

RAYLEIGH NUMBERS .............................................................................................. 29

2.1 INTRODUCTION .................................................................................................... 29

2.2 PHYSICAL PROBLEM............................................................................................. 30

2.3 MODELLING AND IMPLEMENTATION IN OPENFOAM® ......................................... 31

2.4 COMPARISON BETWEEN THE OPENFOAM® CODE AND NUMERICAL BENCHMARKS

.................................................................................................................................. 32

2.5 RESULTS .............................................................................................................. 35

2.5.1 𝑅𝑎𝑒𝑥𝑡 = 106 ......................................................................................................... 36

2.5.2 𝑅𝑎𝑒𝑥𝑡 = 107 ÷ 108 ................................................................................................ 41

2.5.3 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109 ............................................................................................... 51

2.6 EFFECTS OF THE PRANDTL NUMBER ON THE FLOW REGIME ................................ 55

2.7 EFFECTS OF THERMAL INERTIA ........................................................................... 58

2.7.1 Implementation in OpenFOAM® ........................................................................ 58

2.7.2 Results ................................................................................................................. 60

2.8 FINAL REMARKS .................................................................................................. 63

REFERENCES ............................................................................................................. 64

CHAPTER 3: DYNAMIC STABILITY FOR SINGLE-PHASE NATURAL

CIRCULATION IN RECTANGULAR LOOPS ........................................................... 67

3.1 INTRODUCTION .................................................................................................... 67

3.2 EXPERIMENTAL SET-UP AND PROCEDURE ............................................................ 68

3.3 STABILITY MAPS FOR SINGLE-PHASE NATURAL CIRCULATION LOOPS ................... 70

3.4 MODELLING AND IMPLEMENTATION IN OPENFOAM® ......................................... 73

3.4.1 Case 1: Water model in OpenFOAM® ................................................................. 74

3.4.2 Case 2: Pipe Inertia model in OpenFOAM® ....................................................... 75

3.4.3 Case 3: Heat Exchanger model in OpenFOAM® ................................................ 77

3.4.4 Mesh generation and independence of the solution ........................................... 78

3.5 NUMERICAL SIMULATION RESULTS VS. EXPERIMENTAL DATA ............................. 79

3.5.1 Case 1: Water model ........................................................................................... 80

3.5.2 Case 2: Pipe Inertia model .................................................................................. 82

3.5.3 Case 3: Heat Exchanger model ........................................................................... 85

3.5.4 CDF model vs. O-O model ................................................................................... 92

3.6 FINAL REMARKS .................................................................................................. 94

REFERENCES ............................................................................................................. 95

CHAPTER 4: DYNAMIC STABILITY FOR NATURAL CIRCULATION WITH

INTERNALLY HEATED FLUID: THE DYNASTY FACILITY ................................. 99

4.1 INTRODUCTION .................................................................................................... 99

4.2 DESCRIPTION OF THE DYNASTY FACILITY ....................................................... 100

4.2.1 Hitec® molten salt ............................................................................................. 102

CONTENTS

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4.3 STABILITY MAPS FOR SINGLE-PHASE NATURAL CIRCULATION LOOPS WITH

INTERNALLY HEATED FLUID ..................................................................................... 103

4.4 MODELLING AND IMPLEMENTATION IN OPENFOAM® ........................................ 105

4.5 RESULTS ............................................................................................................ 107

4.6 FINAL REMARKS ................................................................................................. 116

REFERENCES ........................................................................................................... 118

CONCLUSION ........................................................................................................... 121

APPENDIX: ASSESSMENT OF THE OPENFOAM® FOR THE CASE OF 2D

CONVENTIONAL NATURAL CIRCULATION IN A DIFFERENTIALLY HEATED

AIR-FILLED SQUARE CAVITY AT DIFFERENT RAYLEIGH NUMBERS ........... 125

A.1 INTRODUCTION .................................................................................................. 125

A.2 MODELLING AND IMPLEMENTATION IN OPENFOAM® ....................................... 126

A.3 RESULTS ............................................................................................................ 127

A.3.1 Laminar flow regime (𝑅𝑎 = 103 ÷ 106) ............................................................. 127

A.3.2 Transition flow regime (𝑅𝑎 = 107 ÷ 108) .......................................................... 132

A.3.3 Turbulent flow regime (𝑅𝑎 = 1.58 × 109) .......................................................... 134

A.4 EFFECTS OF THERMAL INERTIA ......................................................................... 140

A.4.1 Modelling and implementation in OpenFOAM® .............................................. 140

A.4.2 Results ............................................................................................................... 141

A.5 FINAL REMARKS................................................................................................. 144

REFERENCES ........................................................................................................... 145

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ACKNOWLEDGEMENTS

During the period spent working on this thesis work, many people supported

me and helped me in several situations, and for this reasons I am glad to

express my gratitude.

First of all, I wish to thank my supervisor, Prof. Lelio Luzzi, for giving me the

chance to work on this project and the opportunity to enrich my cultural

baggage.

I would thank Prof. Mario Misale for providing the experimental data

required for the validation of the models, giving an important added value to

my work.

I am grateful to all the Nuclear Reactors Group for the fruitful and enjoyable

working environment, showing trustworthiness and professionalism at the

same time. A particular acknowledgment is due to Dr. Matteo Zanetti, for the

very pleasant and stimulating conversations made during my period here.

A very special and heartfelt thanks is direct to my assistant supervisor, Ing.

Alessandro Pini, for his helpfulness in every circumstance. He teaches me a

lot of things, also beyond the topic of my thesis work, succeeding in making

them more interesting. Above all, I would like to thank him for the patience

he showed in following my work, regardless of his countless tasks.

Last but not least, I wish to express my gratitude to my parents, who gave me

the possibility to study far from my hometown, I won’t let you down.

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ABSTRACT

The interest on natural circulation systems derives from the passive safety

they can ensure in removing heat, avoiding the use of active components. It is

necessary to correctly predict the dynamics of natural circulation by means of

a depth knowledge of the phenomena involved, in order to avoid undesired

oscillating behaviours. In this thesis work, several 3D CFD simulations are

run by means of the OpenFOAM® code. Standard solvers are modified in

order to take into account also the effect of Internal Heat Generation (IHG),

in view of the renewed interest in Molten Salt Reactors (MSR) for new

generation nuclear power plants. The solvers are firstly assessed against

buoyancy-driven cavity benchmarks in case of conventional natural

circulation (i.e., without IHG), considering a wide range of Rayleigh number,

and a very good agreement is obtained. A sensitivity study with several

RANS turbulence models is also carried out. Successively, the case of IHG is

considered, highlighting the main effects it induces. In the second part of the

thesis, several models are developed for the study of single-phase natural

circulation loops and are validated, in case of conventional natural

circulation, against the experimental data provided by the L2 facility of

DIME-TEC Labs (Genova University) and available in the open literature.

The effect of thermal inertia is essential in order to correctly reproduce the

dynamic behaviour while a complete model of the heat exchanger allows

predicting the absolute value of the fluid temperature field. At last, the

presence of IHG is introduced, simulating some operative transients of the

DYNASTY facility, currently under construction at Politecnico di Milano. The

results are compared with those obtained through the 1D Object-Oriented

model adopted for the design of the facility. Some difference is observed for

operative transients near the transition curve from stable equilibria to

unstable ones, on the stability maps, but the results can be considered

acceptable.

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SOMMARIO

Per garantire la rimozione del calore, i sistemi a convezione naturale sono dei

buoni candidati, poiché caratterizzati da un elevato grado di sicurezza, data

l’assenza di componenti attivi. Uno studio dei fenomeni in gioco è necessario

per evitare comportamenti oscillanti indesiderati.

In questo lavoro sono state effettuate numerose simulazioni CFD utilizzando

la libreria open source di OpenFOAM®. I solutori standard sono stati inoltre

modificati per considerare anche il caso di fluidi con una sorgente interna di

energia, in vista del rinnovato interesse per i Reattori a Sali Fusi per impianti

nucleari di nuova generazione. I solutori sono stati verificati nel caso della

cavità in circolazione naturale convenzionale (senza generazione interna di

calore). È stato inoltre effettuato uno studio di sensitività utilizzando diversi

modelli di turbolenza. In seguito, è stata introdotta la generazione interna di

calore al fine di studiare le differenze rispetto al caso convenzionale. Nella

seconda parte del lavoro, sono stati sviluppati diversi modelli per lo studio

della circolazione naturale convenzionale in configurazioni a circuito. Tali

modelli sono stati validati utilizzando i dati ottenuti tramite l’apparato

sperimentale L2 del DIME-TEC (Università di Genova) e riportati in

letteratura. In questo caso, è necessario tener conto dell’effetto capacitivo

delle pareti per riprodurre correttamente la dinamica del sistema, mentre un

modello completo dello scambiatore di calore permette di predire il campo di

temperatura del fluido. Infine, il modello è stato esteso al caso con sorgente

interna di energia, prendendo come riferimento il circuito dell’apparato

sperimentale DYNASTY, in costruzione presso il Politecnico di Milano. I

risultati sono stati confrontati con quelli ottenuti con il modello utilizzato per

la progettazione di DYNASTY. Qualche differenza è stata osservata per punti

di lavoro prossimi alla curva di transizione tra equilibri stabili e instabili,

nelle mappe di stabilità, ma complessivamente i risultati sono in buon

accordo.

ESTRATTO IN ITALIANO

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Introduzione

L’uso di sistemi di raffreddamento sempre più affidabili è richiesto in

numerose applicazioni ingegneristiche e in particolare per gli impianti

nucleari, dove la rimozione del calore di decadimento deve essere assicurata

sia durante le normali operazioni di spegnimento sia in quelle incidentali. A

tal proposito, i sistemi passivi a circolazione naturale sono degli ottimi

candidati per garantire un elevato grado di affidabilità, in quanto permettono

di evitare l’uso di elementi attivi, che necessitano di alimentazione elettrica e

possono essere soggetti a guasti. A fronte di questo vantaggio, i sistemi a

convezione naturale richiedono una progettazione particolarmente accurata

in quanto possono essere affetti da oscillazioni dinamiche del campo di moto e

di temperatura. È noto, infatti, che i moti convettivi sono conseguenza

dell’instaurarsi di un equilibrio dinamico tra la forza di galleggiamento,

dovuta a gradienti di densità presenti nel sistema, e le perdite di carico. Tale

equilibrio, tuttavia, può essere sia stabile che instabile.

Nonostante in letteratura siano numerosi gli studi riguardanti la

fenomenologia della circolazione naturale, il caso di fluidi con generazione

interna di calore è un campo poco studiato. Rispetto alla convezione naturale

convenzionale (ossia senza generazione interna di calore), la presenza di una

sorgente interna di energia contribuisce a rendere più complessa la dinamica

del fenomeno (di per sé già non banale), con la possibilità di avere un

comportamento idro-dinamicamente instabile laddove, in caso convenzionale,

il sistema sia caratterizzato da un equilibrio stabile.

Negli ultimi anni, la convezione naturale di fluidi caratterizzati da una

sorgente interna di energia ha suscitato sempre più interesse, soprattutto per

applicazioni in campo nucleare, di cui il più importante esempio è dato dai

reattori di IV generazione a sali fusi con combustibile circolante (GIF, 2014),

in cui il combustibile nucleare è direttamente disciolto in un sale fuso che

funge anche da termovettore. Di conseguenza la generazione di potenza

avviene nel nocciolo in seguito alle reazioni di fissione e lungo il circuito

primario a causa del decadimento dei prodotti di fissione. A seguito dello

spegnimento del reattore, la produzione di calore lungo l’intero sistema può

alterare l’efficacia di un sistema di rimozione del calore a circolazione

naturale, il cui comportamento dinamico deve essere accuratamente

caratterizzato. Per questo motivo, lo sviluppo di appositi modelli e strumenti


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di simulazione, nonché la realizzazione di mirate campagne sperimentali sono

i punti cruciali del progetto europeo SAMOFAR (SAMOFAR, 2015).

In questo ambito, si inserisce il seguente lavoro di tesi. La dinamica della

circolazione naturale è studiata per mezzo di tecniche CFD, considerando

diversi regimi di moto e configurazioni geometriche, sfruttando la libreria a

volumi finiti OpenFOAM®. Due solutori standard del codice sono stati

modificati in modo da tener conto anche della fonte interna di energia.

Il lavoro è così strutturato:

Inizialmente è stato ripreso un benchmark, ampiamente utilizzato in

letteratura, noto come “buoyancy-driven cavity”. In questo modo, è stato

affrontato un primo approccio a OpenFOAM® concentrandosi sull’uso dei

modelli di turbolenza e sul trattamento delle inerzie termiche

(accoppiamento liquido-solido). In particolare, oltre a quello convenzionale,

è stato studiato anche il caso in cui nel sistema sia presente una

generazione interna uniformemente distribuita nel fluido. Questo ha

permesso di mettere in luce alcuni effetti interessanti indotti dalla

generazione interna e dall’uso di fluidi con diversi numeri di Prandtl.

Successivamente, lo studio è stato esteso a una geometria tridimensionale,

considerando i cosiddetti circuiti a circolazione naturale, che sono sistemi

di più pratico interesse per la realizzazione di impianti a convenzione

naturale. Grazie ai dati sperimentali messi a disposizione dal prof. Mario

Misale dell’Università di Genova, è stata condotta una dettagliata

campagna di validazione dei modelli nel caso di un circuito a circolazione

naturale convenzionale, che ha messo in evidenza l’importanza delle

inerzie termiche delle tubazioni del sistema nel definire la dinamica della

circolazione naturale, nonché la necessità di un’accurata scelta del modello

di turbolenza.

Infine, è stato considerato anche il caso di un circuito in cui il fluido è

caratterizzato da una generazione interna di calore, prendendo come

riferimento l’assetto sperimentale di DYNASTY (DYnamics of NAtural

circulation for molten SalT internallY heated) che a breve sarà completato

presso i laboratori del Dipartimento di Energia del Politecnico di Milano. A

questo riguardo, sono stati simulati alcuni transitori operativi con

particolare attenzione alle differenze che si possono riscontrare a livello di

campo di moto e temperatura tra due sistemi geometricamente identici,

differenziati dal fatto che nel primo il riscaldamento distribuito avviene

all’interno del fluido e nel secondo per mezzo di un flusso termico esterno.


xv

Studio numerico della cavità 2D riscaldata dall’esterno, nel

caso di fluidi con generazione interna di energia, a diversi

numeri di Rayleigh

Uno schema rappresentativo del sistema analizzato è riportato in Figura E.1.

Si tratta di una cavità quadrata, alle cui pareti verticali sono imposte due

differenti temperature, mentre quelle orizzontali sono adiabatiche. Il fluido

all’interno della cavità può presentare o meno una sorgente volumetrica di

potenza.

Prima di studiare come la generazione interna di calore possa modificare il

campo di moto all’interno del sistema, i solutori sono stati verificati nel caso

di circolazione naturale convenzionale, usando come confronto il benchmark

numerico fornito da De Vahl Davis (1983), per bassi numeri di Rayleigh

(103 ÷ 106), e quello di Le Quéré (1991), per numeri di Rayleigh (𝑅𝑎 =

𝑔𝛽Δ𝑇𝐿3𝑃𝑟 𝜈2⁄ ) prossimi al valore di transizione (107 ÷ 108), ottenendo ottimi

risultati. L’analisi è stata estesa anche al caso di regime turbolento

utilizzando i dati sperimentali riportati in Ampofo & Karayiannis (2003) e

utilizzando diversi modelli di turbolenza, in modo da investigare quale tra

questi si adatti meglio alla dinamica del sistema in oggetto. In particolare,

sono stati adottati modelli RANS standard a due equazioni: il modello

standard 𝑘 − 휀, il modello 𝑘 − 𝜔 di Wilcox e il modello 𝑆𝑆𝑇 𝑘 − 𝜔. Inoltre sono

state considerate anche diverse strategie per il trattamento a parete delle

grandezze turbolente: no wall function (previo notevole infittimento della

mesh a parete), standard wall function e scalable wall function (adatte per

regimi turbolenti a basso Reynolds). I dettagli dello studio sono riportati in

Appendice, mentre nel Capitolo 1 sono riportati i dettagli riguardanti il

trattamento della turbolenza. Per quanto riguarda i risultati ottenuti per i

modelli di turbolenza, in Figura E.2 è riportato un grafico riassuntivo.

Figura E.1: Schema rappresentativo di una cavità 2D con una differenza di temperatura imposta dall’esterno e una sorgente di calore interna.


xvi

Figura E.2: Risultati ottenuti con diversi modelli di turbolenza, normalizzati rispetto ai dati sperimentali riportati in Ampofo & Karayiannis (2003).

𝑉𝑚𝑎𝑥, 𝑋𝑚𝑎𝑥 e 𝑁𝑢1 2⁄ indicano i valori adimensionalizzati a metà altezza della

cavità del massimo della componente lungo y della velocità, la corrispondente

ascissa ed il valore locale del numero di Nusselt lungo la parete calda,

rispettivamente. I valori riportati sono stati normalizzati rispetto ai valori del

benchmark sperimentale.

Come si può notare dalla Figura E.2, i modelli che più si avvicinano ai dati di

riferimento sono il 𝑘 − 휀 standard con scalable wall functions e il modello

𝑆𝑆𝑇 𝑘 − 𝜔 senza wall functions.

Dopo aver analizzato il caso convenzionale, è stato preso in considerazione

l’effetto della generazione interna di calore, usando come confronto i dati del

benchmark numerico di Shim & Hyun (1997) e ottenendo anche in questo

caso degli ottimi risultati. Un esempio dei risultati ottenuti è riportato in

Figura E.3.

Successivamente, sono state svolte numerose simulazioni considerando una

variazione del numero di Rayleigh che coprisse tutti i regimi di moto. In

particolare è stata posta attenzione agli effetti indotti dalla generazione

interna sul campo di moto e di temperatura. A tale scopo, riprendendo i lavori

di Kawara et al. (1990), Fusegi et al. (1992) e Shim & Hyun (1997), oltre al

numero di Rayleigh convenzionale (da qui in avanti chiamato Rayleigh

esterno) è stato definito il cosiddetto numero di Rayleigh interno (𝑅𝑎𝑖𝑛𝑡 =

𝑔𝛽𝐿5𝑃𝑟𝑞′′′ 𝜈3𝜌𝑐𝑝⁄ ).


xvii

Figura E.3: Confronto tra le linee di contorno della stream function (sinistra) e le distribuzioni di temperatura con isoterme (destra) ottenute in Shim & Hyun (1997)

(sopra) e i risultati ottenuti con OpenFOAM® (sotto) nel caso 𝑅𝑎𝑒𝑥𝑡 = 105 e

𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100.

Al variare del rapporto tra il numero di Rayleigh interno e quello esterno,

l’effetto della sorgente volumetrica di potenza cambia. In particolare, l’effetto

della generazione interna è maggiore all’aumentare di tale rapporto, come

mostrato in Figura E.4 e E.5, nel caso di Rayleigh esterno pari a 107. Indicate

con 𝑇ℎ > 𝑇𝑐 le temperature delle pareti e con 𝑇𝑖𝑛𝑡 la temperatura del fluido,

all’aumentare del Rayleigh interno (a parità di 𝑅𝑎𝑒𝑥𝑡), 𝑇𝑖𝑛𝑡 tende a diventare

maggiore non solo di 𝑇𝑐 ma anche di 𝑇ℎ. Quando 𝑇𝑖𝑛𝑡 > 𝑇ℎ, il moto ascendente

lungo la parete a temperatura 𝑇ℎ si inverte. Accade dunque che lungo

entrambe le pareti il moto sia discendente, ma poiché la differenza di

temperatura tra 𝑇𝑖𝑛𝑡 e 𝑇𝑐 e tra 𝑇𝑖𝑛𝑡 e 𝑇ℎ è diversa, si ha la rottura della

simmetria del campo di moto (Figura E.4(c) e E.5(c)) che in genere

caratterizza questi sistemi1.

Inoltre, se si confrontano i risultati ottenuti per diversi numeri di Rayleigh

esterno, a parità del rapporto 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ , si nota che all’aumentare di 𝑅𝑎𝑒𝑥𝑡,

l’effetto della generazione interna è sempre meno marcato. In particolare, al

crescere del numero di Rayleigh esterno, il fluido si muove sempre più in

prossimità delle pareti della cavità tendendo a recuperare il percorso regolare

a singola cella tipico del caso a circolazione convenzionale.

1 Nel caso di sola generazione interna (𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞), le pareti della cavità sono alla stessa

temperatura (𝑇ℎ − 𝑇𝑐 = 0) e il campo di moto è simmetrico (Figure E.4(d) e E.5(d)).


xviii

(a) (b) (c) (d)

Figura E.4: Confronto delle distribuzioni di velocità nel caso di 𝑅𝑎𝑒𝑥𝑡 = 10

7: (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ =0; (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.

(a) (b) (c) (d)

Figura E.5: Confronto delle distribuzioni di temperatura nel caso di 𝑅𝑎𝑒𝑥𝑡 = 10

7: (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.

Per visualizzare meglio il concetto, nelle Figure E.6 e E.7 è riportato il

confronto tra i profili di velocità e temperatura per i diversi numeri di

Rayeigh considerati.

(a) (b) (c) (d)

Figura E.6: Confronto delle distribuzioni di velocità nel caso di 𝑅𝑎𝑒𝑥𝑡 = 10

6(a), 𝑅𝑎𝑒𝑥𝑡 = 107(b),

𝑅𝑎𝑒𝑥𝑡 = 108 (c) e 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 10

9 (d), a parità di rapporto 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100.


xix

(a) (b) (c) (d)

Figura E.7: Confronto delle distribuzioni di temperatura nel caso di 𝑅𝑎𝑒𝑥𝑡 = 10

6(a), 𝑅𝑎𝑒𝑥𝑡 =107(b), 𝑅𝑎𝑒𝑥𝑡 = 10

8 (c) e 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109 (d), a parità di rapporto 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100.

Inoltre, anche l’effetto del numero di Prandtl del fluido è stato analizzato, in

modo qualitativo, usando come riferimento i lavori di Fusegi et al. (1992) e di

Arcidiacono et al. (2001). Un esempio è riportato in Figura E.8, confrontando i

risultati ottenuti con due diversi numeri di Prandtl, a parità di numero di

Grashof, nel caso di sola generazione interna di calore.

(a) (b)

(c) (d)

Figura E.8: Confronto tra il campo di velocità (sinistra) e temperatura (destra) ottenuti con un numero di Grashof interno pari a 109, nel caso di 𝑃𝑟 = 0.0321 (a e b)

(Arcidiacono et al., 2001) e 𝑃𝑟 = 0.71 (c e d) (OpenFOAM®).

Il medesimo studio è stato successivamente ripetuto considerando la presenza

di pareti solide ai lati verticali della cavità, al fine di introdurre lo studio di

sistemi caratterizzati da regioni mutiple (accoppiamento liquido-solido) per il

quale OpenFOAM® offre un solutore ad hoc). Mentre per alcuni sistemi la

presenza delle inerzie termiche del solido altera la dinamica della circolazione


xx

naturale (per esempio, nei circuiti a circolazione naturale), nel caso della

cavità considerata non è indotto alcun effetto perché le pareti sono a

temperatura imposta. I risultati numerici hanno verificato a posteriori questo

fatto.

Riassumendo, i principali effetti osservati sono stati:

Il peso relativo della generazione interna di calore, rispetto al caso

convenzionale di calore ceduto dall’esterno, diventa sempre maggiore

all’aumentare del rapporto 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ , ma allo stesso tempo diventa

sempre meno marcato all’aumentare del numero di Raylegh esterno,

mantenendo fisso il rapporto 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ ;

Per numeri di Rayleigh esterno prossimi al valore di transizione tra

regime laminare a regime turbolento, l’effetto aggiuntivo della generazione

esterna può portare a modificare la dinamica del sistema;

Se si considera un fluido caratterizzato da un numero di Prandtl

sufficientemente diverso, si è osservato che, nel caso di sola generazione

interna (𝑅𝑎𝑒𝑥𝑡 = 0) e a parità di numero di Grashof interno, il

comportamento dinamico del sistema può cambiare anche di molto,

passando da un moto ordinato e regolare a uno caotico;

Introducendo l’effetto delle inerzie termiche delle pareti, non sono state

osservate variazioni significative delle principali grandezze del fluido

rispetto al caso senza inerzie.

Studio del comportamento dinamico dei circuiti a

circolazione naturale convenzionale

Completata l’analisi della cavità, lo studio del comportamento dinamico della

circolazione naturale è stato esteso al caso di un sistema a circolazione

naturale a circuito, assumendo come geometria di riferimento quella

dell’assetto sperimentale L2, sito presso i laboratori del DIME-TEC

(Università di Genova). Uno schema del circuito è riportato in Figura E.9,

mentre le principali grandezze geometriche sono riassunte in Tabella E.1.

Dettagli sull’assetto sperimentale possono essere trovati nel documento

tecnico IAEA (2014) e in Sezione 3.2.


xxi

Figura E.9: Schema semplificato del circuito L2.

Tabella E.1: Principali grandezze geometriche del circuito L2

𝑫 (𝒎𝒎) 𝑾 (𝒎𝒎) 𝑯 (𝒎𝒎) 𝑳𝒉 (𝒎𝒎) 𝑳𝒄 (𝒎𝒎) 𝑳𝒕𝒐𝒕 (𝒎𝒎) 𝑳𝒕 𝑫⁄ 𝑯 𝑾⁄

30 1112 988 960 900 4100 136.67 0.88

I risultati ottenuti sono stati quindi confrontati con i dati sperimentali

provenienti dal medesimo circuito L2 e gentilmente forniti dal Prof. Mario

Misale. Tali dati sono stati il frutto di una campagna sperimentale volta ad

investigare come la temperatura del pozzo freddo possa influenzare la

dinamica del sistema.

A causa dell’elevato costo computazionale, sono stati considerati solo due casi,

particolarmente significativi, tra quelli a disposizione: il 4 ℃ ed il 18 ℃ per la

temperatura del pozzo freddo. Diversi regimi di moto sono inoltre stati

considerati in modo da verificare come cambiassero i risultati, utilizzando per

il regime turbolento il modello 𝑆𝑆𝑇 𝑘 − 𝜔 con scalable wall function.

Tre differenti modelli sono stati testati, con diversi livelli di dettaglio, in modo

da approssimare sempre meglio l’assetto sperimentale e costatare quale

livello di dettaglio sia necessario per riuscire a prevedere i comportamenti

sperimentalmente osservati.

Nel primo modello è stato trascurato l’effetto delle inerzie termiche,

simulando soltanto la regione del fluido. Nonostante l’intervallo di

temperatura previsto fosse tra i limiti osservati per entrambe le temperature,

nel caso a 18 ℃ il comportamento dinamico non è stato correttamente

simulato. Per quanto riguarda il regime di moto, invece, nel caso a 4 ℃ è stato

osservato che i risultati ottenuti con il regime di moto turbolento

approssimano meglio i dati sperimentali.


xxii

Nel secondo modello è stato introdotto lo spessore delle pareti dei tubi, mentre

il pozzo freddo (cooler) è stato modellizzato come una temperatura imposta

all’esterno della parete. Ciò, da un lato, ha permesso di cogliere correttamente

la dinamica del sistema grazie alla presenza delle pareti, ma, dall’altro lato,

la modellizzazione del cooler ha portato a sottostimare il campo di

temperatura del fluido.

Per ultimo, il pozzo freddo è stato modellato considerando il sistema completo

del cilindro coassiale alla sezione del cooler con una portata imposta. Una

leggera sottostima delle temperature continua a persistere, ma i risultati sono

complessivamente accettabili. Inoltre sono stati osservati anche interessanti

effetti 3D come la formazione di zone (plug) calde e fredde, all’origine della

dinamica nei circuiti in convezione naturale, e le distribuzioni radiali di

temperatura e velocità, di cui un esempio è dato nelle Figure E.10÷E.13.

Figura E.10: Distribuzione della temperatura a 2430 𝑠. Inversione della portata dal senso orario ad antiorario.

Figura E.11: Distribuzione della

componente lungo 𝑥 della velocità a 2430 𝑠. Moto antiorario.

Figura E.12: Distribuzione radiale della temperatura a 1990 𝑠 in una sezione

della gamba inferiore.

Figura E.13: Distribuzione radiale della

velocità a 1990 𝑠 in una sezione della gamba inferiore.


xxiii

Per quest’ultimo modello è stato inoltre considerato anche un secondo modello

di turbolenza, il Realizable 𝑘 − 휀, in modo da osservare come tale scelta

influenza i risultati delle simulazioni. Con il secondo modello, però, non è

stato possibile prevedere correttamente la dinamica del sistema.

Il comportamento dinamico ottenuto con i diversi modelli è stato infine

confrontato con le informazioni fornite dalle mappe di stabilità, ottenute da

un approccio semi-analitico, sia nel caso con che senza inerzie termiche. Il

vantaggio di usare le mappe di stabilità è che queste permettono di prevedere

l’equilibrio asintotico cui perviene il sistema in modo relativamente semplice,

ma non permetto di studiare l’evoluzione nel tempo con cui tale equilibrio è

raggiunto. È stata riscontrata una buona coerenza tra le informazioni delle

mappe di stabilità e la dinamica predetta dalle simulazioni CFD. Infine, i

risultati ottenuti con le simulazioni CFD sono stati confrontati anche con i

risultati di un modello Object-Oriented (O-O) 1D semplificato sviluppato dal

Gruppo Reattori Nucleari del Politecnico di Milano, osservando un buon

accordo tra i modelli numerici e i dati sperimentali. Informazioni su tale

modello possono essere trovate in Cammi et al. (2016). La differenza di

temperatura ottenuta con i due modelli (CFD e O-O) e il confronto con i dati

sperimentali sono riportati per i due casi considerati (4 ℃ e 18 ℃) in Figura

E.14 ed E.15.

Figura E.14: Dati sperimentali e dati numerici (O-O e CFD con modello di tubolenza 𝑆𝑆𝑇 𝑘 − 𝜔 e scalable wall

function). Caso 4 ℃.

Figura E.15: Dati sperimentali e dati numerici (O-O e CFD con modello di tubolenza 𝑆𝑆𝑇 𝑘 − 𝜔 e scalable wall

function). Caso 18 ℃.


xxiv

Studio del comportamento dinamico dei circuiti a

circolazione naturale in presenza di una sorgente interna di

calore: il circuito di DYNASTY

Sulla base dei risultati ottenuti dalla validazione dei modelli nel caso di

circuiti a circolazione naturale convenzionale, si è esteso lo studio al caso in

cui il fluido all’interno del circuito sia caratterizzato da una sorgente interna

di calore. A tal fine, le simulazioni sono state condotte prendendo come

riferimento il circuito di DYNASTY, in costruzione presso i laboratori del

Dipartimento di Energia del Politecnico di Milano. La progettazione di questo

assetto sperimentale è stato basato su un modello 1D O-O sviluppato in lavori

precedenti (Pini et al., 2014; Ruiz et al., 2015; Pini et al., 2016; Cammi et al.,

2016). Uno schema del circuito è riportato in Figura E.16, mentre le principali

grandezze geometriche sono riassunte in Tabella E.2. Ulteriori informazioni

sull’assetto sperimentale e sulle proprietà del sale fuso scelto come fluido di

lavoro sono riportate in Sezione 4.2.

Figura E.16: Schema di DYNASTY (non in scala).

Tabella E.2: Principali grandezze geometriche di DYNASTY

𝑫 (𝒎𝒎) 𝝉 (𝒎𝒎) 𝑾 (𝒎𝒎) 𝑯 (𝒎𝒎) 𝑳𝒕𝒐𝒕 (𝒎𝒎) 𝑳𝒕 𝑫⁄ 𝑯 𝑾⁄

38.2 2 2400 3200 11200 293.19 1.33


xxv

Lo scopo di questa sezione è dunque quello di simulare alcuni transitori di

DYNASTY utilizzando il modello 3D CFD con inerzie precedentemente

validato. L’andamento nel tempo della portata in massa ottenuta nelle

diverse simulazioni è stato confrontato con i risulti corrispondenti del modello

1D O-O. In particolare, sono stati considerati tre diversi livelli di potenza

termica fornita al sistema (10 𝑘𝑊, 2 𝑘𝑊 e 0.5 𝑘𝑊) e due diverse modalità per

fornire tale potenza, ossia un flusso termico imposto dall’esterno lungo tutto il

tubo del circuito, eccetto che nella sezione del cooler, oltre che al caso di

sorgente interna di calore. Questa scelta deriva dalle problematiche legate

alla realizzazione della generazione interna in un fluido, dal punto di vista

sperimentale. Nel caso di circuiti con un rapporto lunghezza-su-diametro di

molto maggiore all’unità è già stato evidenziato numericamente che queste

due modalità sono equivalenti (Cammi et al., 2016).

In generale, la dinamica nel caso di flusso termico esterno risulta essere,

almeno nel transitorio iniziale, più lenta rispetto al caso con generazione

interna, mentre il valore assoluto della portata è leggermente superiore. Il

primo effetto può essere spiegato prendendo in considerazione come l’energia

termica sia trasferita al fluido nei due casi. Nel caso di flusso termico

dall’esterno, il trasferimento di energia può riassumersi in tre fasi principali:

l’energia è inizialmente assorbita alle pareti del tubo e spesa per aumentarne

la temperatura; in seguito, l’energia è trasferita agli strati di fluido in

prossimità della parete, lasciando la parte del core a una temperatura

inferiore; nella terza ed ultima fase, il calore penetra sempre più in profondità

sino ad avere una distribuzione quasi uniforme in direzione radiale. Nel caso

di generazione interna, invece, il fluido è già caratterizzato sin dai primi

istanti da una distribuzione uniforme di energia, per cui le tre fasi riportate

in precedenza non hanno luogo. La minor portata è dovuta al fatto che parte

dell’energia termica del fluido è spesa per riscaldare la parete, fenomeno che

invece non ha luogo nel caso di flusso esterno imposto. Ciò porta ad avere una

distribuzione di temperatura leggermente inferiore e, di conseguenza, una

portata minore.

Le simulazioni con il modello CFD hanno quindi fornito un’ulteriore prova a

sostegno dell’equivalenza tra le due diverse modalità di riscaldamento,

informazione utile dal punto di vista sperimentale in quanto permette di

evitare tutta una serie di problematiche legate alla realizzazione di una

sorgente interna di energia.

Per quanto riguarda il confronto dei risultati tra i due modelli numerici,

complessivamente è stato osservato un buon accordo. I risultati possono così

riassumersi:


xxvi

Per il caso a 10 𝑘𝑊 è stata predetta correttamente la dinamica del

sistema, osservando un errore relativo tra i dati delle simulazioni CFD e

quelli del modello O-O non superiori all’8 %;

Nel caso a 2 𝑘𝑊, il comportamento dinamico osservato è coerente con

quello predetto dalle mappe di stabilità, con la differenza che nel modello

O-O la portata è caratterizzata da oscillazioni unidirezionali, mentre nel

modello CFD sono state osservate anche inversioni nella direzione del

flusso;

Per il caso a 0.5 𝑘𝑊, i due modelli predicono una diversa dinamica. Questa

discrepanza può essere spiegata dal fatto che il punto rappresentativo di

tale transitorio nella mappa di stabilità è in prossimità della curva di

transizione tra equilibrio instabile e stabile, per cui gli effetti 3D come

anche la diversa modellizzazione dello scambio termico e delle perdite di

carico possono influenzare in modo significativo i risultati.

In Figura E.17 è riportato come esempio il confronto tra il transitorio ottenuto

con un flusso termico imposto dall’esterno e quello nel caso di generazione

interna, nel caso a 10 𝑘𝑊, mentre nelle Figure E.18÷E.21 sono mostrati i

profili radiali di temperatura e velocità all’equilibrio, in due diverse sezioni

del circuito. Infine, nelle Figure E.22÷E.24 è mostrato il confronto tra

l’approccio CFD e il modello O-O per i tre casi considerati.

Figura E.17: Confronto dell’andamento nel tempo della portata in massa ottenuto con OpenFOAM® nel caso di un flusso termico imposto dall’esterno (A-EHF) e di

generazione interna (IHG), per una potenza di 10 𝑘𝑊.


xxvii

(a) (b)

Figura E.18: Distribuzione di temperatura all’equilibrio in corrispondenza del cooler (a) e della gamba inferiore del circuito (b) nel caso di un flusso termico imposto, per

una potenza di 10 𝑘𝑊.

(a) (b)

Figura E.19: Distribuzione della velocità all’equilibrio in corrispondenza del cooler (a) e della gamba inferiore del circuito (b) nel caso di un flusso termico imposto, per


(a) (b)

Figura E.20: Distribuzione di temperatura all’equilibrio in corrispondenza del cooler (a) e della gamba inferiore del circuito (b) nel caso di generazione interna, per una

potenza di 10 𝑘𝑊.


xxviii

(a) (b)

Figura E.21: Distribuzione della velocità all’equilibrio in corrispondenza del cooler (a) e della gamba inferiore del circuito (b) nel caso di generazione interna, per una

potenza di 10 𝑘𝑊.

(a)

(b)

Figura E.22: Confronto tra i risultati ottenuti con l’approccio CFD e il modello O-O, nel caso di generazione interna (a) e di flusso termico imposto dall’esterno (b), per



xxix

(a)

(b)



Conclusioni

In questo lavoro di tesi, il fenomeno della circolazione naturale è stato

studiato numericamente adottando un approccio 3D CFD. In particolare,

solutori standard della libreria a volumi finiti OpenFOAM® sono stati

modificati in modo da poter tener conto anche dell’eventuale presenza di

sorgenti interne di energia. L’interesse per questo genere di sistemi non è

soltanto di tipo scientifico, ma deriva anche dalle possibili applicazioni

pratiche, di cui un esempio rilevante è dato dai Reattori a Sali Fusi per

impianti nucleari di nuova generazione.

Dapprima è stato presentato il caso classico di una cavità 2D con una

differenza di temperatura imposta alle pareti dall’esterno, di cui sono

disponibili numerosi benchmark in letteratura, in modo da prendere

dimestichezza con il codice con un caso ampiamente studiato in letteratura.

Sono stati considerati inoltre diversi regimi di moto, facendo variare il

numero di Rayleigh in un’ampia finestra di valori.


xxx

(a)

(b)


una potenza di 0.5 𝑘𝑊.

Ottimi risultati sono stati ottenuti nel caso di flusso laminare e di regime di

moto in transizione, mentre da un’analisi di sensitività sulla scelta del

modello di turbolenza RANS, nel caso di regime turbolento, è stato osservato

che i migliori risultati si ottengono utilizzando il modello standard 𝑘 − 휀 con

scalabe wall function oppure il modello 𝑆𝑆𝑇 𝑘 − 𝜔 di Menter senza wall

function.

Successivamente, lo studio è stato esteso anche al caso combinato di calore

fornito dall’esterno e sorgente interna di energia, evidenziando i principali

effetti che quest’ultimo induce sia sulla dinamica del sistema che sul moto del

fluido in generale. Tra gli effetti più interessanti si evidenzia il fatto che, per

numeri di Rayleigh prossimi al valor critico di transizione da regime di moto

laminare a turbolento nel caso convenzionale (ossia senza sorgente interna di

calore), la presenza della generazione interna può incidere in modo tale da

modificare la dinamica del sistema, passando da un moto regolare ad uno

caotico, fenomeno influenzato anche dal numero di Prandtl del fluido stesso.


xxxi

Infine, è stata considerata la presenza delle inerzie termiche delle pareti

solide per introdurre sistemi caratterizzati da più regioni come i circuiti a

circolazione naturale.

La validazione è stata condotta poi nel caso di più pratico interesse di un

circuito a circolazione naturale, confrontando le simulazioni effettuate con tre

diversi modelli, nel caso convenzionale, con i dati forniti dell’assetto

sperimentale L2 del DIME-TEC (Università di Genova). I migliori risultati,

sia in termine di dinamica del sistema che di distribuzione di temperatura del

fluido, sono stati ottenuti includendo l’effetto capacitivo delle pareti dei tubi e

considerando un modello completo per lo scambiatore di calore, verificando

anche che il modello 𝑆𝑆𝑇 𝑘 − 𝜔 permette meglio di simulare il comportamento

turbolento del fluido all’interno del circuito.

È stato infine simulato il caso in cui il fluido all’interno del circuito è

caratterizzato da una generazione interna di energia, prendendo come

riferimento il circuito di DYNASTY, in costruzione presso il Politecnico di

Milano. In particolare, sono stati considerati tre diversi transitori di potenza:

10 𝑘𝑊, 2 𝑘𝑊 e 0.5 𝑘𝑊.

Oltre al caso di una sorgente volumetrica uniformemente distribuita, è stato

considerato anche quello in cui la potenza è fornita al sistema per mezzo di un

flusso termico imposto dall’esterno e distribuito uniformemente lungo tutto il

circuito, eccetto la sezione del cooler, in modo da evidenziare se con queste

due modalità sia possibile pervenire al medesimo risultato. Le simulazioni

hanno confermato che i risultati sono molto simili tra loro, almeno nel caso di

circuiti caratterizzati da un rapporto lunghezza-su-diametro molto maggiore

dell’unità. I risultati ottenuti sono stati poi confrontati con quelli del modello

1D O-O utilizzato per la progettazione dell’apparato sperimentale, osservando

qualche differenza per valori di potenza tali per cui il corrispondente punto di

lavoro è prossimo alla linea di transizione tra equilibrio stabile ed instabile,

nella mappa di stabilità. Ciò deriva principalmente dal fatto che con un

approccio CFD non è necessario ricorrere a correlazioni empiriche per

modellare lo scambio termico e le perdite di carico, oltre al fatto che è

possibile tener conto anche di effetti 3D.

Scopo di lavori futuri sarà quello di migliorare il modello CFD, adottando

modelli di turbolenza avanzati, e di validare i modelli per mezzo dei dati

sperimentali ottenuti dal circuito di DYNASTY, una volta ultimato.

xxxiii

NOMENCLATURE

List of symbols

𝑎 Coefficient of thermo-physical polynomial dependence on the temperature

𝐴 Surface (𝑚2)

𝑏 Generic constant term vector

𝑐 Specific heat (𝐽 𝑘𝑔−1𝐾−1)

𝐷 Diameter (𝑚)

𝐸 Energy spectrum of turbulence (𝐽)

𝑓 Darcy friction factor

𝐹 Blending Function

𝑔 Gravity acceleration (𝑚 𝑠−2)

𝐺 Mass flux (𝑘𝑔 𝑚−2𝑠−1)

𝐺𝑟 Grashof number

ℎ Convective heat transfer coefficient (𝑊 𝑚−2𝐾−1)

𝐻 Height (𝑚)

𝑘 Turbulent kinetic energy for unit mass (𝑚2 𝑠−2)

𝐾 Mean flow kinetic energy for unit mass (𝑚2 𝑠−2)

𝐿 Length (𝑚)

𝑀 Generic matrix coefficient

𝑁𝑢 Nusselt number

𝑝 Pressure (𝑃𝑎)

𝑃 Mean flow pressure (𝑃𝑎)

𝑃𝑟 Prandtl number

𝑞′′ Localized heat flux (𝑊 𝑚−2)

𝑞#′′ Uniformly distributed external heat flux (𝑊 𝑚−2)

𝑞′′′ Internal distributed volumetric heat source (𝑊 𝑚−3)

𝑅 Conductive thermal resistance of the pipe (𝑚2𝐾 𝑊−1)

𝑅𝑎 Rayleigh number

𝑅𝑒 Reynolds number

𝑠 Curvilinear axial coordinate (𝑚)

𝑆 Pipe cross-section (𝑚2)

𝑆𝑡 Stanton number

𝑡 Time (𝑠)

𝑇 Temperature (𝐾 or ℃)

𝒖 Velocity vector (𝑚 𝑠−1)

𝑼 Mean flow velocity vector (𝑚 𝑠−1)

𝑢 X-velocity component (𝑚 𝑠−1)

𝑈 Dimensionless x-velocity component

𝑣 Y-velocity component (𝑚 𝑠−1)

𝑉 Dimensionless y-velocity component

𝑊 Width (𝑚)

𝑥, 𝑦 Cartesian coordinate (𝑚)

NOMENCLATURE

xxxiv

𝑋, 𝑌 Dimensionless Cartesian coordinate

Special characters

𝛼 Thermal diffusivity (𝑚2𝑠)

𝛽 Thermal expansion coefficient (𝐾−1)

𝒷 Dimensionless constant in the auxiliary condition for specific dissipation

Γ Mass flow rate (𝑘𝑔 𝑠−1)

𝛿 Perturbation

𝒟 Diffusion coefficient for a generic scalar quantity (𝑚2𝑠−1)

𝜃 Dimensionless temperature

ℐ Turbulent intensity

𝜅 Wavenumber ( 𝑚−1)

𝜆 Thermal conductivity (𝑊 𝑚−1𝐾−1)

ℓ Eddies length scale (𝑚)

𝜇 Dynamic viscosity (𝑃𝑎 𝑠)

𝜈 Momentum diffusivity ( 𝑚2𝑠)

𝜉 Generic coordinate

𝓅 Perturbation pulsation (𝑠−1)

𝑝𝑒𝑓𝑓 Effective pressure (𝑃𝑎)

𝜌 Density (𝑘𝑔 𝑚−3)

ℛ(𝓅) Real part of the perturbation pulsation (𝑠−1)

𝜎 Turbulent Prandtl number

𝜏 Viscous stress (𝑃𝑎)

𝓉 Pipe thickness (𝑚)

𝜙, 𝜓 Generic flow variable

Φ, Ψ Mean flow generic flow variable

�� Spatial-dependent part of the perturbed generic flow variable

𝜔 Specific dissipation (𝑠−1)

𝒮 Source term of the generic scalar quantity

𝑠 Turbulent deformation tensor

𝑆 Mean flow deformation tensor

�� Length of the infinitesimal shell of the pipe (𝑚)

�� Lateral surface of the infinitesimal shell of the pipe (𝑚2)

𝒰 Eddies velocity scale (𝑚 𝑠−1) 𝒱 Volume (𝑚3)

�� Volume of the infinitesimal shell of the pipe (𝑚3)

𝑦+ Dimensionless wall distance for a wall-bounded flow

Subscripts-superscripts

0 Steady-state value

1 2⁄ Mid-height

∗ Reference value

NOMENCLATURE

xxxv

′ Fluctuating component

~ Weighted average

𝑐 Cold/cooler

𝑒𝑥𝑡 External

𝑓 Fluid

ℎ Hot/heater

𝑖 Inner shell of the pipe

𝑖𝑛 Inlet

𝑖𝑛𝑡 Internal

𝑚 Modified

𝑚𝑎𝑥 Maximum

𝑚𝑖𝑛 Minimum

𝑜 Outer shell of the pipe

𝑜𝑢𝑡 Outlet

𝑡 Turbulent

𝑡𝑜𝑡 Total

𝑤 Wall

Acronyms

1-2-3D One-Two-Three Dimensional

A-EHF All-External Heat Flux

𝐴𝑟𝑧 Horizontal Aspect Ratio

BC Boundary Condition

CFD Computational Fluid Dynamics

DIME-

TEC

Department of Mechanics, Energetics, Management and Transportation

- Thermal Engineering and Environmental Conditioning

DNS Direct Numerical Simulation

DYNASTY DYnamics of NAtural circulation for molten SalT internallyY heated

FR Flow Regime

FVM Finite Volume Method

GUI Graphic User Interface

HE Heat Exchanger

HHHC Horizontal-Heater Horizontal-Cooler

HR High-Reynolds

IAEA International Atomic Energy Agency

IC Initial Condition

IHG Internal Heat Generation

LES Large Eddies Simulation

LHF Localized Heat Flux

LR Low-Reynolds

MSR Molten Salt Reactor

O-O Object-Oriented

RANS Reynolds-Average Navier-Stokes equations

RE Relative Error

SAMOFAR SAfety Assessment of the MOlten salt FAst Reactor

NOMENCLATURE

xxxvi

SST Shear Stress Transport

TC Thermo-Couple

TI Thermal Inertia

VHHC Vertical-Heater Horizontal-Cooler

xxxix

LIST OF FIGURES

FIGURA E.1: SCHEMA RAPPRESENTATIVO DI UNA CAVITÀ 2D CON UNA DIFFERENZA DI

TEMPERATURA IMPOSTA DALL’ESTERNO ED UNA SORGENTE DI CALORE INTERNA. XV

FIGURA E.2: RISULTATI OTTENUTI CON DIVERSI MODELLI DI TURBOLENZA,

NORMALIZZATI RISPETTO AI DATI SPERIMENTALI RIPORTATI IN AMPOFO &

KARAYIANNIS (2003). ........................................................................................... XVI

FIGURA E.3: CONFRONTO TRA LE LINEE DI CONTORNO DELLA STREAM FUNCTION

(SINISTRA) E LE DISTRIBUZIONI DI TEMPERATURA CON ISOTERME (DESTRA)

OTTENUTE IN SHIM & HYUN (1997) (SOPRA) E I RISULTATI OTTENUTI CON

OPENFOAM® (SOTTO) NEL CASO 𝑅𝑎𝑒𝑥𝑡 = 105 E 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100. ................... XVII

FIGURA E.4: CONFRONTO DELLE DISTRIBUZIONI DI VELOCITÀ NEL CASO DI 𝑅𝑎𝑒𝑥𝑡 = 107:

(A) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; (D)

𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ................................................................................................ XVIII

FIGURA E.5: CONFRONTO DELLE DISTRIBUZIONI DI TEMPERATURA NEL CASO DI 𝑅𝑎𝑒𝑥𝑡 =

107: (A) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; (D)

𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ................................................................................................ XVIII

FIGURA E.6: CONFRONTO DELLE DISTRIBUZIONI DI VELOCITÀ NEL CASO DI 𝑅𝑎𝑒𝑥𝑡 =

106(A), 𝑅𝑎𝑒𝑥𝑡 = 107(B), 𝑅𝑎𝑒𝑥𝑡 = 10

8 (C) E 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109 (D), A PARITÀ DI

RAPPORTO 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100. ........................................................................... XVIII

FIGURA E.7: CONFRONTO DELLE DISTRIBUZIONI DI TEMPERATURA NEL CASO DI 𝑅𝑎𝑒𝑥𝑡 =

106(A), 𝑅𝑎𝑒𝑥𝑡 = 107(B), 𝑅𝑎𝑒𝑥𝑡 = 10

8 (C) E 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109 (D), A PARITÀ DI

RAPPORTO 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100. ............................................................................. XIX

FIGURA E.8: CONFRONTO TRA IL CAMPO DI VELOCITÀ (SINISTRA) E TEMPERATURA

(DESTRA) OTTENUTI CON UN NUMERO DI GRASHOF INTERNO PARI A 109, NEL CASO

DI 𝑃𝑟 = 0.0321 (A E B) (ARCIDIACONO ET AL., 2001) E 𝑃𝑟 = 0.71 (C E D)

(OPENFOAM®). .................................................................................................... XIX

FIGURA E.9: SCHEMA SEMPLIFICATO DEL CIRCUITO L2............................................... XXI

FIGURA E.10: DISTRIBUZIONE DELLA TEMPERATURA A 2430 𝑠. INVERSIONE DELLA

PORTATA DAL SENSO ORARIO AD ANTIORARIO. .................................................... XXII

FIGURA E.11: DISTRIBUZIONE DELLA COMPONENTE LUNGO 𝑥 DELLA VELOCITÀ A 2430 𝑠.

MOTO ANTIORARIO. ............................................................................................. XXII

FIGURA E.12: DISTRIBUZIONE RADIALE DELLA TEMPERATURA A 1990 𝑠 IN UNA SEZIONE

DELLA GAMBA INFERIORE. .................................................................................. XXII

FIGURA E.13: DISTRIBUZIONE RADIALE DELLA VELOCITÀ A 1990 𝑠 IN UNA SEZIONE

DELLA GAMBA INFERIORE. .................................................................................. XXII

FIGURA E.14: DATI SPERIMENTALI E DATI NUMERICI (O-O E CFD CON MODELLO DI

TUBOLENZA 𝑆𝑆𝑇 𝑘 − 𝜔 E SCALABLE WALL FUNCTION). CASO 4 ℃. ...................... XXIII

FIGURA E.15: DATI SPERIMENTALI E DATI NUMERICI (O-O E CFD CON MODELLO DI

TUBOLENZA 𝑆𝑆𝑇 𝑘 − 𝜔 E SCALABLE WALL FUNCTION). CASO 18 ℃. .................... XXIII

FIGURA E.16: SCHEMA DI DYNASTY (NON IN SCALA). .............................................. XXIV

FIGURA E.17: CONFRONTO DELL’ANDAMENTO NEL TEMPO DELLA PORTATA IN MASSA

OTTENUTO CON OPENFOAM® NEL CASO DI UN FLUSSO TERMICO IMPOSTO

LIST OF FIGURES

xl

DALL’ESTERNO (A-EHF) E DI GENERAZIONE INTERNA (IHG), PER UNA POTENZA DI

10 𝑘𝑊. ................................................................................................................ XXVI

FIGURA E.18: DISTRIBUZIONE DI TEMPERATURA ALL’EQUILIBRIO IN CORRISPONDENZA

DEL COOLER (A) E DELLA GAMBA INFERIORE DEL CIRCUITO (B) NEL CASO DI UN

FLUSSO TERMICO IMPOSTO, PER UNA POTENZA DI 10 𝑘𝑊. ................................. XXVII

FIGURA E.19: DISTRIBUZIONE DELLA VELOCITÀ ALL’EQUILIBRIO IN CORRISPONDENZA

DEL COOLER (A) E DELLA GAMBA INFERIORE DEL CIRCUITO (B) NEL CASO DI UN

FLUSSO TERMICO IMPOSTO, PER UNA POTENZA DI 10 𝑘𝑊. ................................. XXVII

FIGURA E.20: DISTRIBUZIONE DI TEMPERATURA ALL’EQUILIBRIO IN CORRISPONDENZA

DEL COOLER (A) E DELLA GAMBA INFERIORE DEL CIRCUITO (B) NEL CASO DI

GENERAZIONE INTERNA, PER UNA POTENZA DI 10 𝑘𝑊. ...................................... XXVII

FIGURA E.21: DISTRIBUZIONE DELLA VELOCITÀ ALL’EQUILIBRIO IN CORRISPONDENZA

DEL COOLER (A) E DELLA GAMBA INFERIORE DEL CIRCUITO (B) NEL CASO DI

GENERAZIONE INTERNA, PER UNA POTENZA DI 10 𝑘𝑊. .................................... XXVIII

FIGURA E.22: CONFRONTO TRA I RISULTATI OTTENUTI CON L’APPROCCIO CFD E IL

MODELLO O-O, NEL CASO DI GENERAZIONE INTERNA (A) E DI FLUSSO TERMICO

IMPOSTO DALL’ESTERNO (B), PER UNA POTENZA DI 10 𝑘𝑊. .............................. XXVIII



IMPOSTO DALL’ESTERNO (B), PER UNA POTENZA DI 2 𝑘𝑊. ................................... XXIX



IMPOSTO DALL’ESTERNO (B), PER UNA POTENZA DI 0.5 𝑘𝑊. ................................. XXX

FIGURE 1.1: ENERGY SPECTRUM OF TURBULENCE (VERSTEEG & MALALASEKERA, 2007).

.............................................................................................................................. 14

FIGURE 1.2: EXAMPLE OF DIRECTORY STRUCTURE. ...................................................... 25

FIGURE 2.1: DIFFERENTIALLY HEATED 2D CAVITY WITH INTERNAL ENERGY SOURCE. . 33

FIGURE 2.2: CASE 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 10: PLOTS OF STREAM FUNCTION AND ISOTHERMS

(SHIM & HYUN, 1997). .......................................................................................... 34

FIGURE 2.3: CASE 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 10: CONTOUR LINES OBTAINED WITH OPENFOAM®. 34

FIGURE 2.4: CASE 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100: PLOTS OF STREAM FUNCTION AND ISOTHERMS

(SHIM & HYUN, 1997). .......................................................................................... 34

FIGURE 2.5: CASE 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100: CONTOUR LINES OBTAINED WITH OPENFOAM®.

.............................................................................................................................. 34

FIGURE 2.6: COMPARISON OF THE VELOCITY DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 106: CASE (A)

𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; CASE (D)

𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ................................................................................................... 40

FIGURE 2.7: COMPARISON OF THE TEMPERATURE DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 106: CASE

(A) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; CASE

(D) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. .............................................................................................. 40



𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ................................................................................................... 45

LIST OF FIGURES

xli



(D) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ..............................................................................................45



𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ....................................................................................................48



(D) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ..............................................................................................48

FIGURE 2.12: PLOT OVER TIME AND MEAN VALUE OF DIMENSIONLESS V-VELOCITY

COMPONENT (𝑅𝑎𝑒𝑥𝑡 = 108, 𝑅𝑎𝑖𝑛𝑡 = 10

10). ...............................................................49

FIGURE 2.13: PLOT OVER TIME AND MEAN VALUE OF DIMENSIONLESS TEMPERATURE

(𝑅𝑎𝑒𝑥𝑡 = 108, 𝑅𝑎𝑖𝑛𝑡 = 10

10 ). ..........................................................................................50

FIGURE 2.14: PLOT OVER TIME AND MEAN VALUE OF DIMENSIONLESS V-VELOCITY

COMPONENT (𝑅𝑎𝑖𝑛𝑡 = 1.58 × 109). ..........................................................................52

FIGURE 2.15: COMPARISON OF THE VELOCITY DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109:

CASE (A) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100;

CASE (D) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. .....................................................................................53

FIGURE 2.16: COMPARISON OF THE TEMPERATURE DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 1.58 ×

109: CASE (A) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ =

100; CASE (D) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ..............................................................................54

FIGURE 2.17: CASE 𝐺𝑟 = 105. ........................................................................................55

FIGURE 2.18: CASE 𝐺𝑟 = 3 × 107. ..................................................................................55

FIGURE 2.19: CASE 𝐺𝑟 = 5.4 × 107. ................................................................................56

FIGURE 2.20: CASE 𝐺𝑟 = 108. ........................................................................................56

FIGURE 2.21: COMPARISON OF THE VELOCITY DISTRIBUTION: (A) 𝐺𝑟𝑖𝑛𝑡 = 107, (B) 𝐺𝑟𝑖𝑛𝑡 =

108, (C) 𝐺𝑟𝑖𝑛𝑡 = 109, ARCIDIACONO ET AL. (2001); (D) 𝐺𝑟𝑖𝑛𝑡~1.4 × 10

7, (E)

𝐺𝑟𝑖𝑛𝑡~1.4 × 108, (F) 𝐺𝑟𝑖𝑛𝑡~2.2 × 10

9, OPENFOAM®. ...............................................57

FIGURE 2.22: COMPARISON OF THE TEMPERATURE DISTRIBUTION: (A) 𝐺𝑟𝑖𝑛𝑡 = 107, (B)

𝐺𝑟𝑖𝑛𝑡 = 108, (C) 𝐺𝑟𝑖𝑛𝑡 = 10

9, ARCIDIACONO ET AL. (2001); (D) 𝐺𝑟𝑖𝑛𝑡~1.4 × 107, (E)

𝐺𝑟𝑖𝑛𝑡~1.4 × 108, (F) 𝐺𝑟𝑖𝑛𝑡~2.2 × 10

9, OPENFOAM®. ...............................................57

FIGURE 3.1: L2 EXPERIMENTAL RIG PICTURE (IAEA, 2014)...........................................68

FIGURE 3.2: L2 EXPERIMENTAL FACILITY SCHEME. .......................................................68

FIGURE 3.3: SKETCH OF THE POSITION OF THE MAIN THERMOCOUPLES (MISALE &

GARIBALDI, 2010). .................................................................................................69

FIGURE 3.4: DISCRETIZATION OF THE PIPE WALLS (A) AND ELECTRICAL EQUIVALENT

MODEL (B). .............................................................................................................72

FIGURE 3.5: STABILITY MAP FOR THE L2 FACILITY WITH AND WITHOUT THE EFFECT OF

THE PIPING MATERIALS THERMAL PROPERTIES (TI EFFECTS). ...............................73

FIGURE 3.6: COMPARISON BETWEEN THE SYSTEM EQUILIBRIA REFERRED TO THE

DIFFERENT EXPERIMENTAL CASES AND THE STABILITY MAPS. ...............................73

FIGURE 3.7: T-JUNCTION FOR THE EXPANSION TANK. ....................................................77

FIGURE 3.8: DETAILS OF THE ADOPTED COMPUTATIONAL GRID. ....................................78

FIGURE 3.9: COMPUTATIONAL GRID IN THE HEAT-EXCHANGER ZONE. ...........................78

LIST OF FIGURES

xlii

FIGURE 3.10: TEMPERATURE SIGNALS FROM THERMOCOUPLE TC-5 AND TC-20 FOR THE

EXPERIMENTS 1A, 1B, AND 1C. ............................................................................... 79

FIGURE 3.11: TEMPERATURE SIGNALS FROM THERMOCOUPLE TC-5 AND TC-20 FOR THE

EXPERIMENT 9. ...................................................................................................... 80

FIGURE 3.12: EXPERIMENTAL VS. SIMULATED TEMPERATURE DIFFERENCE ACROSS THE

COOLING SECTION FOR THE EXPERIMENT 1A OBTAINED NEGLECTING THE PIPE

WALL INERTIA AND USING THE LAMINAR MODEL. .................................................. 80


COOLING SECTION FOR THE EXPERIMENT 1A OBTAINED NEGLECTING THE PIPE

WALL INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. ........................... 81

FIGURE 3.14: EXPERIMENTAL VS. SIMULATED TEMPERATURE TAKEN AT PROBE POSITION

TC-5 (TOP) AND TC-20 (BOTTOM) FOR THE EXPERIMENT 1A OBTAINED NEGLECTING

THE PIPE WALL INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. ............ 81


COOLING SECTION FOR THE EXPERIMENT 9 OBTAINED NEGLECTING THE PIPE WALL

INERTIA AND USING THE LAMINAR MODEL. ............................................................ 82


COOLING SECTION FOR THE EXPERIMENT 9 OBTAINED NEGLECTING THE PIPE WALL

INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. .................................... 82


COOLING SECTION FOR THE EXPERIMENT 1A OBTAINED CONSIDERING THE PIPE

WALL INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. .......................... 83


TC-5 (TOP) AND TC-20 (BOTTOM) FOR THE EXPERIMENT 1A OBTAINED

CONSIDERING THE PIPE WALL INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE

MODEL. .................................................................................................................. 83


COOLING SECTION FOR THE EXPERIMENT 9 OBTAINED CONSIDERING THE PIPE

WALL INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. .......................... 84


TC-5 (TOP) AND TC-20 (BOTTOM) FOR THE EXPERIMENT 9 OBTAINED CONSIDERING

THE PIPE WALL INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. ............ 84

FIGURE 3.21: SIMULATED TEMPERATURE DIFFERENCE ACROSS THE COOLING SECTION

FOR THE EXPERIMENT 9 WITH AND WITHOUT THE TI EFFECT OBTAINED USING THE

𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. ........................................................................... 85

FIGURE 3.22: SIMULATED TEMPERATURE TAKEN AT PROBE POSITION TC-5 (TOP) AND

TC-20 (BOTTOM) FOR THE EXPERIMENT 9 WITH AND WITHOUT TI EFFECT OBTAINED

USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. .......................................................... 85


COOLING SECTION FOR THE EXPERIMENT 1A OBTAINED WITH HE MODEL AND

USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. ......................................................... 86


TC-5 (TOP) AND TC-20 (BOTTOM) FOR THE EXPERIMENT 1A OBTAINED WITH HE

MODEL AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. ...................................... 86

LIST OF FIGURES

xliii


COOLING SECTION FOR THE EXPERIMENT 9 OBTAINED WITH HE MODEL AND USING

THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. ....................................................................87


TC-5 (TOP) AND TC-20 (BOTTOM) FOR THE EXPERIMENT 9 OBTAINED WITH HE

MODEL AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. .......................................87

FIGURE 3.27: CFD TEMPERATURE DISTRIBUTION AT 1970 𝑠, HOT (COLD) PLUG IS

DEVELOPING. .........................................................................................................89

FIGURE 3.28: CFD TEMPERATURE DISTRIBUTION AT 1990 𝑠, SLIDING OF THE HOT (COLD)

PLUG IN THE RIGHT (LEFT) VERTICAL LEG. .............................................................89

FIGURE 3.29: CFD TEMPERATURE DISTRIBUTION AT 2000 𝑠, THE HOT (COLD) PLUG

OCCUPIES THE ENTIRE RIGHT (LEFT) VERTICAL LEG. .............................................89

FIGURE 3.30: CFD TEMPERATURE DISTRIBUTION AT 2010 𝑠, THE HOT (COLD) PLUG

STARTS TO ENTER IN THE LEFT (RIGHT) VERTICAL LEG. .........................................89

FIGURE 3.31: CFD TEMPERATURE DISTRIBUTION AT 2330 𝑠, MASS FLOW RATE

INVERSION FROM THE ANTICLOCKWISE DIRECTION TO THE CLOCKWISE ONE. .......90

FIGURE 3.32: CFD TEMPERATURE DISTRIBUTION AT 2430 𝑠, MASS FLOW RATE

INVERSION FROM THE CLOCKWISE DIRECTION TO THE ANTICLOCKWISE ONE. .......90

FIGURE 3.33: DISTRIBUTION OF THE X COMPONENT OF THE VELOCITY (𝑈𝑥) AT 2330 𝑠

(CLOCKWISE CIRCULATION). ...................................................................................90

FIGURE 3.34: DISTRIBUTION OF THE X COMPONENT OF THE VELOCITY (𝑈𝑥) AT 2430 𝑠

(ANTICLOCKWISE CIRCULATION). ...........................................................................90

FIGURE 3.35: CFD RADIAL TEMPERATURE DISTRIBUTION TAKEN AT TC-20 POSITION

(1990 𝑠). .................................................................................................................91

FIGURE 3.36: CFD RADIAL TEMPERATURE DISTRIBUTION TAKEN AT TC-10 POSITION

(1990 𝑠). .................................................................................................................91

FIGURE 3.37: CFD RADIAL VELOCITY MAGNITUDE DISTRIBUTION TAKEN AT TC-20

POSITION (1990 𝑠). ..................................................................................................91

FIGURE 3.38: CFD RADIAL VELOCITY MAGNITUDE DISTRIBUTION TAKEN AT TC-10

POSITION (1990 𝑠). ..................................................................................................91


COOLING SECTION FOR THE EXPERIMENT 1A OBTAINED WITH HE MODEL AND

USING THE REALIZABLE 𝑘 − 휀 TURBULENCE MODEL. ..............................................92


COOLING SECTION FOR THE EXPERIMENT 9 OBTAINED WITH HE MODEL AND USING

THE REALIZABLE 𝑘 − 휀 TURBULENCE MODEL. ........................................................92

FIGURE 3.41: EXPERIMENTAL VS. SIMULATED TEMPERATURE (O-O AND CFD WITH

𝑆𝑆𝑇 𝑘 − 𝜔) TAKEN AT PROBE POSITION TC-20 FOR EXPERIMENT 1A. ......................93


𝑆𝑆𝑇 𝑘 − 𝜔) TAKEN AT PROBE POSITION TC-5 FOR EXPERIMENT 1A. ........................93


𝑆𝑆𝑇 𝑘 − 𝜔) TAKEN AT PROBE POSITION TC-20 FOR EXPERIMENT 9. ........................93


𝑆𝑆𝑇 𝑘 − 𝜔) TAKEN AT PROBE POSITION TC-5 FOR EXPERIMENT 9. ..........................93

LIST OF FIGURES

xliv

FIGURE 3.45: EXPERIMENTAL VS. SIMULATED (O-O AND CFD WITH 𝑘 − 𝜔 )

TEMPERATURE DIFFERENCE ACROSS THE COOLING SECTION FOR THE EXPERIMENT

1A. ......................................................................................................................... 93

FIGURE 3.46: EXPERIMENTAL VS. SIMULATED (O-O AND CFD WITH 𝑆𝑆𝑇 𝑘 − 𝜔)

TEMPERATURE DIFFERENCE ACROSS THE COOLING SECTION FOR THE EXPERIMENT

9. ........................................................................................................................... 93

FIGURE 4.1: DYNASTY SCHEME (NOT IN SCALE)........................................................ 100

FIGURE 4.2: HITEC® DENSITY (TOP) AND DYNAMIC VISCOSITY (BOTTOM) AS A FUNCTION

OF TEMPERATURE. ............................................................................................... 103

FIGURE 4.3: DYNASTY STABILITY MAP IN THE 𝑆𝑡𝑚-𝐺𝑟𝑚 PLANE FOR THE IHG, A-EHF

AND LHF CASES. ................................................................................................. 104

FIGURE 4.4: DETAILS OF THE ADOPTED COMPUTATIONAL GRID. ................................. 105

FIGURE 4.5: COMPARISON OF THE ASYMPTOTIC EQUILIBRIA OBTAINED WITH THE 1D O-

O MODEL AND THE STABILITY MAP, FOR THE OPERATIVE TRANSIENTS CONSIDERED.

............................................................................................................................ 107

FIGURE 4.6: MASS FLOW RATE TRANSIENT BEHAVIOUR OBTAINED WITH THE

OPENFOAM® CODE FOR THE IHG AND A-EHF CASES, AT 10 𝑘𝑊. ....................... 108

FIGURE 4.7: CFD VS. O-O MODELS FOR THE IHG (A) AND A-EHF (B) HEATING MODES,

AT 10 𝑘𝑊. ............................................................................................................ 108

FIGURE 4.8: STEADY-STATE RADIAL TEMPERATURE DISTRIBUTION TAKEN AT THE

COOLER SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF A-EHF AND 10 𝑘𝑊.

............................................................................................................................ 109

FIGURE 4.9: STEADY-STATE RADIAL VELOCITY DISTRIBUTION TAKEN AT THE COOLER

SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF A-EHF AND 10 𝑘𝑊. .......... 109


COOLER SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF IHG AND 10 𝑘𝑊. . 110


SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF IHG AND 10 𝑘𝑊. .............. 110


OPENFOAM® CODE FOR THE IHG AND A-EHF CASES, AT 2 𝑘𝑊. ......................... 111


AT 2 𝑘𝑊. .............................................................................................................. 111

FIGURE 4.14: RADIAL DISTRIBUTION OF TEMPERATURE (A) AND VELOCITY (B) TAKEN AT

THE RIGHT VERTICAL LEG AT 500 𝑠. ..................................................................... 112

FIGURE 4.15: RADIAL TEMPERATURE DISTRIBUTION TAKEN AT THE COOLER SECTION (A)

AND AT THE BOTTOM LEG (B) IN CASE OF A-EHF, AT 1430 𝑠 (MAXIMUM OF THE

OSCILLATION) AND 2 𝑘𝑊. ..................................................................................... 112

FIGURE 4.16: RADIAL VELOCITY DISTRIBUTION TAKEN AT THE COOLER SECTION (A) AND

AT THE BOTTOM LEG (B) IN CASE OF A-EHF, AT 1430 𝑠 (MAXIMUM OF THE


FIGURE 4.17: RADIAL TEMPERATURE DISTRIBUTION TAKEN AT THE COOLER SECTION (A)

AND AT THE BOTTOM LEG (B) IN CASE OF IHG, AT 1660 𝑠 (MAXIMUM OF THE


LIST OF FIGURES

xlv

FIGURE 4.18: RADIAL VELOCITY DISTRIBUTION TAKEN AT THE COOLER SECTION (A) AND

AT THE BOTTOM LEG (B) IN CASE OF IHG, AT 1660 𝑠 (MAXIMUM OF THE



OPENFOAM® CODE FOR THE IHG AND A-EHF CASES, AT 0.5 𝑘𝑊. ....................... 113


AT 0.5 𝑘𝑊. ............................................................................................................ 114

FIGURE 4.20: MASS FLOW RATE TRANSIENT BEHAVIOUR OBTAINED WITH THE 1D O-O

MODEL FOR THE IHG AND A-EHF CASES, AT 0.4 𝑘𝑊. ........................................... 114


COOLER SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF A-EHF AND 0.5 𝑘𝑊.

............................................................................................................................. 115


SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF A-EHF AND 0.5 𝑘𝑊. .......... 115


COOLER SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF IHG AND 0.5 𝑘𝑊. . 115


SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF IHG AND 0.5 𝑘𝑊. .............. 116

FIGURE A.1: DIFFERENTIALLY HEATED 2D CAVITY. .................................................... 126

FIGURE A.2: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 103. .................. 128






FIGURE A.8: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 1.58 × 109 WITH

LOW-REYNOLDS 𝑘 − 휀 MODEL. ............................................................................. 139

FIGURE A.9: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 1.58 × 109 WITH

THE 𝑆𝑆𝑇 𝑘 − 𝜔 MODEL. ......................................................................................... 139

xlvii

LIST OF TABLES

TABELLA E.1: PRINCIPALI GRANDEZZE GEOMETRICHE DEL CIRCUITO L2 .................... XXI

TABELLA E.2: PRINCIPALI GRANDEZZE GEOMETRICHE DI DYNASTY ....................... XXIV

TABLE 2.1: THERMAL IC AND BC, SIZE OF THE CAVITY, THERMAL PROPERTIES AT 300 K

(BERGMAN ET AL., 2011) ........................................................................................32

TABLE 2.2: COMPARISON WITH BENCHMARK .................................................................33

TABLE 2.3: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1..............................................................37

TABLE 2.4: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100 ..........................................................38

TABLE 2.5: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞ ............................................................39

TABLE 2.6: COMPARISON 𝑅𝑎 = 106 ................................................................................39

TABLE 2.7: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1..............................................................42

TABLE 2.8: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100 ..........................................................43

TABLE 2.9: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞ ............................................................44

TABLE 2.10: COMPARISON 𝑅𝑎 = 107 ..............................................................................44

TABLE 2.11: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1 ............................................................46

TABLE 2.12: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100 ........................................................46

TABLE 2.13: GRID SENSITIVITY 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞...........................................................47

TABLE 2.14: COMPARISON 𝑅𝑎 = 108 ..............................................................................47

TABLE 2.15: NEW SETTING FOR THE CASE AT 𝑅𝑎 = 1.58 × 109 ......................................51

TABLE 2.16: LAMINAR VS. TURBULENT FLOW REGIMES (CAVITY ( 0.1 × 0.1)) .................52

TABLE 2.17: LAMINAR VS. TURBULENT FLOW REGIMES (CAVITY ( 0.75 × 0.75)) .............53

TABLE 2.18: COMPARISON 𝑅𝑎 = 1.58 × 109( 𝑆𝑆𝑇 𝑘 − 𝜔, GRID (310 × 310)) ....................53

TABLE 2.19: EFFECT OF THE PRANDTL NUMBER ON THE FLOW REGIME (FR) ................57

TABLE 2.20: THERMO-PHYSICAL PROPERTIES (300 K) ..................................................59

TABLE 2.21: GRID SENSITIVITY FOR THE TI CASE ( 𝑅𝑎𝑒𝑥𝑡 = 106, 𝑅𝑎𝑖𝑛𝑡 = 10

8) ................60

TABLE 2.22: GRID SENSITIVITY FOR THE TI CASE ( 𝑅𝑎𝑒𝑥𝑡 = 0, 𝑅𝑎𝑖𝑛𝑡 = 106) ...................61

TABLE 2.23: TI RESULTS (𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109, 𝑅𝑎𝑖𝑛𝑡 = 1.58 × 10

11) ...............................61

TABLE 2.24: TI RESULTS ( 𝑅𝑎𝑒𝑥𝑡 = 0, 𝑅𝑎𝑖𝑛𝑡 = 1.58 × 109) ..............................................62

TABLE 2.25: COMPARISON WITH PREVIOUS DATA (LAMINAR FLOW REGIME) .................62

TABLE 2.26: COMPARISON WITH PREVIOUS DATA (TURBULENT FLOW REGIME) ............63

TABLE 3.1: L2 EXPERIMENTAL RIG DIMENSIONS ............................................................69

TABLE 3.2: EXPERIMENTAL CAMPAIGN TEST MATRIX .....................................................70

TABLE 3.3: THERMAL PROPERTIES OF THE WATER AT 300 𝐾 (BERGMAN ET AL., 2011) ..75

TABLE 3.4: INITIAL AND BOUNDARY CONDITION ADOPTED FOR THE WATER MODEL ......75

TABLE 3.5: THERMO-PHYSICAL PROPERTIES (THE TEMPERATURE 𝑇 IS EXPRESSED IN

KELVIN). ................................................................................................................76

TABLE 3.6: INLET/OUTLET BOUNDARY CONDITIONS ADOPTED FOR THE HE ..................77

TABLE 3.7: MESH SENSITIVITY ......................................................................................78

TABLE 3.8: STEADY-STATE VALUES OF TEMPERATURE ...................................................84

TABLE 3.9: MEAN VALUES OF TEMPERATURE IN THE TIME INTERVAL 2030 𝑠-2400 𝑠 ......86

LIST OF TABLES

xlviii

TABLE 3.10: STEADY-STATE VALUES OF TEMPERATURE ................................................ 87

TABLE 4.1: DYNASTY DIMENSIONS ........................................................................... 101

TABLE 4.2: MESH SENSITIVITY ................................................................................... 105

TABLE 4.3: THERMO-PHYSICAL PROPERTIES (THE TEMPERATURE 𝑇 IS EXPRESSED IN

KELVIN) ............................................................................................................... 106

TABLE 4.4: SUMMARY OF THE RESULTS OBTAINED FOR THREE DIFFERENT TRANSIENTS

OF THE DYNASTY FACILITY LOOP ...................................................................... 117

TABLE A.1: THERMAL IC AND BC, SIZE OF THE CAVITY, THERMAL PROPERTIES AT 300 K

(BERGMAN ET AL., 2011) ..................................................................................... 126

TABLE A.2: GRID SENSITIVITY 𝑅𝑎 = 103 ..................................................................... 128




TABLE A.6: COMPARISON WITH BENCHMARKS ............................................................ 131



TABLE A.9: COMPARISON WITH BENCHMARKS ............................................................ 133

TABLE A.10: GRID SENSITIVITY: STANDARD 𝑘 − 휀 MODEL (HI-REYNOLDS WALL

FUNCTION)........................................................................................................... 135

TABLE A.11: GRID SENSITIVITY: STANDARD 𝑘 − 휀 MODEL (LOW-REYNOLDS WALL

FUNCTION)........................................................................................................... 136

TABLE A.12: GRID SENSITIVITY: WILCOX 𝑘 − 𝜔 MODEL (HI-REYNOLDS WALL FUNCTION)

............................................................................................................................ 136

TABLE A.13: GRID SENSITIVITY: WILCOX 𝑘 − 𝜔 MODEL (LOW-REYNOLDS WALL

FUNCTION)........................................................................................................... 136

TABLE A.14: GRID SENSITIVITY: MENTER 𝑆𝑆𝑇 𝑘 − 𝜔 MODEL (HI-REYNOLDS WALL

FUNCTION)........................................................................................................... 136

TABLE A.15: GRID SENSITIVITY: MENTER 𝑆𝑆𝑇 𝑘 − 𝜔 MODEL (LOW-REYNOLDS WALL

FUNCTION)........................................................................................................... 137

TABLE A.16: COMPARISON WITH BENCHMARK ............................................................ 137

TABLE A.17: MENTER 𝑆𝑆𝑇 𝑘 − 𝜔 MODEL (NON-UNIFORM GRID) .................................. 139

TABLE A.18: THERMO-PHYSICAL PROPERTIES (300 𝐾) ............................................... 141

TABLE A.19: GRID SENSITIVITY FOR THE TI CASE (𝑅𝑎 = 106) ...................................... 142

TABLE A.20: GRID SENSITIVITY FOR THE TI CASE (𝑅𝑎 = 1.58 × 109, 𝑘 − 휀 MODEL) ..... 142

TABLE A.21: TI RESULTS (𝑅𝑎 = 1.58 × 109, 𝑆𝑆𝑇 𝑘 − 𝜔 MODEL) .................................... 142

TABLE A.22: COMPARISON WITH PREVIOUS DATA (𝑅𝑎 = 106) ...................................... 143

TABLE A.23: COMPARISON WITH PREVIOUS DATA (𝑅𝑎 = 1.58 × 109, 𝑘 − 휀 MODEL) ..... 143

3

INTRODUCTION

In nuclear power plants and in engineering applications, the development of

highly reliable emergency passive systems is necessary in order to guarantee

the heat removal in both operative and accidental conditions. In this regard,

growing attention is paid to natural circulation, whose main advantage is to

avoid the use of active components, ensuring a high level of intrinsic safety

and reducing maintenance and operating costs. However, natural circulation

can be characterized by oscillations that can lead to undesired dynamic

behaviours, a drawback that can be avoided thanks to an accurate design of

the system.

In general, natural convection phenomena take place in many applications of

interest, such as the design of energy efficient buildings (Korpela et al., 1982),

cooling of electronic devices, solar collector (Buchberg et al., 1976) and

nuclear reactor cooling. For this reason, natural circulation has been widely

studied both experimentally and numerically. Unlike forced convection, in

which the driving force is external to the fluid, in natural circulation the flow

is due to the simultaneous presence of density gradients and body forces. This

combination gives rise to a buoyancy force, which is the driving force of the

free convection. Variation of the density field can be induced by temperature

or concentration gradients, while the most common body force is the

gravitational one. Since natural circulation is characterized by a very complex

dynamics, many studies have been conducted on simple configurations,

among which the most commonly used is the buoyancy-driven cavity. It

represents a suitable means for experimental studies, thanks to the simplicity

of the experimental set-up and the measurement of the quantities of interest.

Numerical benchmarks widely adopted for the validations of numerical codes

are those provided by De Vahl Davis (1983) and Le Quéré (1991). A general

literature review about buoyancy-cavity can be found in Ostrach (1988).

Natural convection phenomena are made even more complex by the presence

of internal energy sources. Actually, internally heated fluids recently have

gained attention for advanced engineering applications, especially in the

nuclear energy field. In this regard, the main example is given by the

Generation IV Molten Salt Reactor (MSR), in which the nuclear fuel is

directly dissolved in a molten salt that also acts as coolant. As a consequence,

the heat is generated on one hand by fission reactions in core and on the other

hand by decay of the fission products in the primary circuit (GIF, 2014; Serp

et al., 2016). The presence of Internal Heat Generation (IHG) affects the

dynamic behaviour of natural circulation with respect to the conventional

INTRODUCTION

4

case (i.e., without IHG) and can lead to the transition from a stable

equilibrium state to an unstable one. For this reason, the prediction of the

transient-behaviour in such cases is essential in order to avoid hydro-dynamic

instabilities and therefore to ensure a higher degree of safety, which

represents one of the main purposes of Generation IV nuclear reactors. The

theoretical and experimental studies of such effects are some of the objectives

of the SAMOFAR Project (SAMOFAR, 2015).

In studying natural circulation phenomena, two dimensionless numbers are

generally considered, namely the Prandtl number and the Grashof number,

whose combined effects are summarized in the Rayleigh number. Most of the

works on natural convection with internal energy sources deals with cases at

low Rayleigh number and only internal heat generation, rarely including the

effect of thermal inertia. Bergholz (1980) solves the boundary-layer equations

in a closed cavity when internally heated fluids are considered, useful to

obtain the equations valid in the near-wall region and in the core of the

enclosure. Acharya & Goldstein (1985) conducts a numerical study on natural

convection in a differentially heated square cavity and internal energy

sources with different aspect ratios and inclination, with an external Rayleigh

number in the range (103 ÷ 106) and an internal Rayleigh number in the

range (104 ÷ 107), while an experimental study is reported in Lee & Goldstein

(1988). Experiments on a fluid with a Prandtl number of ~5 can be found in

Kawara et al. (1990). Fusegi et al. (1992 a, b) conduct numerical study on

square and rectangular enclosures. Shim & Hyun (1997) present a numerical

study on transient behaviour of a square cavity heated both externally and

internally with an external Rayleigh number in the range (105 ÷ 107), an

internal Rayleigh number in the range (106 ÷ 107), and a Prandtl number of

the order of 1. Natural convection with higher Rayleigh numbers and a

Prandtl number of ~7 is conducted by Liaqat & Baytas (2000), taking into

account also the effect of thermal inertia (Liaqat & Baytas, 2001). Low

Prandtl number natural convection in enclosures with volumetric sources and

different aspect ratios is numerically studied by Di Piazza & Ciofalo (2000),

Arcidiacono et al. (2001) and Arcidiacono & Ciofalo (2001).

The buoyancy-driven cavity is surely a useful system for the study of free

convection, but a more realistic configuration, commonly adopted for natural

circulation systems in engineering applications, is represented by closed

rectangular or toroidal loops. Several approaches have been developed for the

study of such systems, namely semi-analytical linear stability analysis, which

allows defining the asymptotic behaviour of natural circulation loops by

means of dimensionless stability maps, and numerical codes, for simulating

the time-dependent behaviour. Furthermore, also the effect of several

parameters on the stability has been investigated. However, almost all the

works in literature never take into account the effect of IHG.

INTRODUCTION

5

The first theoretical studies were carried out by Keller (1966) and Welander

(1967). In these works, the loop is modelled with a single point heat source

and sink connected by two adiabatic legs. Keller (1966) predicted

unidirectional oscillation and Welander (1967) explained such behaviour as

the growth in time and amplitude of small perturbations, which can produce

an inversion of the flow direction. Successively, Chen (1985) showed the effect

of the aspect ratio on the stability of natural circulation loops. In Vijayan et

al. (2001, 2002, 2007, 2008), the effect of heater and cooler orientations and of

the length-to-diameter ratio on the stability are experimentally investigated,

while in Devia & Misale (2012) the effect of the heat sink temperature is

investigated. Numerical studies have been conducted by Vijayan et al. (1995,

2007), Ambrosini et al. (1998, 2000, 2004), Pilkhwal et al. (2007), Swapnalee

& Vijayan (2011), adopting several approaches (one-dimensional finite

difference method, finite element method and three-dimensional CFD). The

effect of thermal inertia on the dynamics of natural circulation loops has been

less investigated. In Misale et al. (2000), two-dimensional numerical

simulation is reported, while in Misale et al. (2005), the interaction between

several piping materials and the fluid is experimentally studied.

First works on IHG effects are those of Pini et al. (2014) and Ruiz et al.

(2015), in which a linear stability analysis is carried out neglecting the effect

of mass flow oscillation on the convective heat transfer coefficient.

Successively, in Pini et al. (2016), the heat exchange effect on the dynamics is

investigated and several numerical approaches are adopted for solving the

non-linear equations, comparing the results with the information provided by

the stability maps. In Cammi et al. (2016), the studies of the previous works

are extended including also the effect of piping materials showing how

thermal inertia can deeply affect the transient behaviour of natural

circulation loops. The methods developed in these works, however, are purely

methodological. In this regard, the building of the DYNASTY (DYnamics of

NAtural circulation for molten SalT internallY heated) testing facility will be

completed soon at Energy Labs of Politecnico di Milano in order to provide the

necessary experimental data for their validation.

The purpose of this work is to provide three-dimensional CFD models for

simulating natural circulation dynamics in both conventional (i.e., without

IHG) and unconventional natural circulation cases, considering different flow

regimes and taking into account the conjugate heat transfer between solid

wall and fluid region. The thesis is organized as follows.

In the first part, two standard solvers of the OpenFOAM® code

(OpenFOAM®, 2016), namely, the buoyantBoussinesqPimpleFoam and the

chtMultiRegionFoam, are modified in order to take into account also the

effect of internal heat source. The results obtained with the modified solvers

are compared with numerical benchmarks available in literature in the

INTRODUCTION

6

simple case of the buoyancy-driven cavity with and without the presence of

internal heat generation, adopting several turbulence models in order to

familiarize with the structure of the code. Successively, the simultaneous

presence of external and internal heating is considered in a wide range of

Rayleigh numbers, in particular in the laminar-to-turbulent transition flow

regime region, in order to investigate the modifications induced by the

presence of internal energy source. Some consideration is also reported for

what concern the effect of the Prandtl number in case of IHG.

In the second part of the thesis, the modified numerical solvers and the

developed models are validated in case of a single-phase natural circulation

loop without IHG, using the experimental data provided by the L2 facility of

DIME-TEC Labs (Genova University) and available from IAEA (2014),

focusing on the dynamic behaviour of the system and highlighting three-

dimensional effects in this kind of systems. The obtained results are

compared with the information provided by the semi-analytical approach

developed in Cammi et al. (2016). The same code is used for simulating some

operative transients of the DYNASTY facility for the case of a natural

circulation loop with IHG, comparing the results with the one-dimensional

numerical solutions reported in Cammi et al. (2016).

Final remarks and future works are reported at the end of the thesis.

References

Acharya, S. & Goldstein, R. J., 1985. Natural convection in an externally

heated vertical or inclined square box containing internal energy sources.

ASME Journal of Heat Transfer, 107(4), pp. 855-866.

Ambrosini, W. & Ferreri, J., 1998. The effect of truncation error on numerical

prediction of linear stability boundaries in a natural circulation single-phase

loop. Nuclear Engineering and Design, 183(1), p. 53–76.

Ambrosini, W. & Ferreri, J., 2000. Stability analysis of single phase

thermosiphons loops by finite-difference numerical methods. Nuclear

Engineering and Design, 201(1), pp. 11-23.

Ambrosini, W., Forgione, N., Ferreri, J. & Bucci, M., 2004. The effect of wall

friction in single-phase natural circulation stability at the transition between

laminar and turbulent flow. Annals of Nuclear Energy, 31(16), pp. 1833-1865.

Ampofo, F. & Karayiannis, T. G., 2003. Experimental benchmark data for

turbulent natural convection in an air filled square cavity. International

Journal of Heat and Mass Transfer, 46(19), pp. 3551-3572.

INTRODUCTION

7

Arcidiacono, S. & Ciofalo, M., 2001. Low-Prandtl number natural convection

in volumetrically heated rectangular enclosures III. Swallow cavity, AR=0.25.

International Journal of Heat and Mass Transfer, 44(16), pp. 3053-3065.

Arcidiacono, S., Di Piazza, I. & Ciofalo, M., 2001. Low-Prandtl number

natural convection in volumetrically heated rectangular enclosures II. Square

cavity, AR=1. International Journal of Heat Transfer, 44(3), pp. 537-550.

Bergholz, R. F., 1980. Natural convection of a heat generating fluid in a closed

cavity. ASME Journal of Heat Transfer, 102(2), pp. 242-247.

Buchberg, H., Catton, I. & Edward, D. K., 1976. Natural convection in

enclosed spaces - A review of application to solar energy collection. ASME

Journal of Heat Transfer, 98(2), pp. 182-188.

Cammi, A., Luzzi, L. & Pini, A., 2016. The influence of the wall thermal

inertia over a single-phase natural convection loop with internally heated

fluids. Chemical Engineering Science (submitted).

Chen, K., 1985. On the instability of closed-loop thermosiphons. Journal of

Heat Transfer, 107(4), pp. 826-832.

De Vahl Davis, G., 1983. Natural convection of air in a square cavity: a bench

mark numerical solution. International Journal for Numerical Methods in

Fluids, 3(3), pp. 249-264.

Devia, F. & Misale, M., 2012. Analysis of the effects of heat sink temperature

on single-phase natural circulation loops behaviour. International Journal of

Thermal Sciences, Volume 59, p. 195–202.

Di Piazza, I. & Ciofalo, M., 2000. Low-Prandtl number natural convection in

volumetrically heated rectangular enclosures I: Slender cavity, AR=4.


Fusegi, T., Hyum, J. N. & Kuwahara, K., 1992. Natural convectionin a

differentially heated square cavity with internal heat generation. Numerical

Heat Transfer, Part A, 21(2), pp. 215-229.

Fusegi, T., Hyun, J. M. & Kuwahara, K., 1992. Numerical study of natural

convection in a differentially heated cavity with internal heat generation.

ASME International Journal of Heat Transfer, 144(3), pp. 773-777.

IAEA, 2014. Progress in Methodologies for the Assessment of Passive System

Reliability in Advanced Reactors. Vienna, IAEA-TECDOC-1752.

INTRODUCTION

8

Kawara, Z., Kishiguci, I., Aoki, N. & Michiyoshi, I., 1990. Natural convection

in a vertical fluid layer with internal heating. Proceeding 27th National Heat

Transfer Symposium, Japan, Volume II, pp. 115-117.

Keller, J. B., 1966. Periodic oscillations in a model of thermal convection.

Journal of Fluid Mechanics, 1(26), pp. 599-606.

Korpela, S. A., Lee, Y. & Drummond, J. E., 1982. Heat transfer through a

double pane window. ASME Journal of Heat Transfer, 104(3), pp. 530-544.

Le Quéré, P., 1991. Accurate solutions to the square thermally heated driven

cavity at high Rayleigh. Computers & Fluids, 20(1), pp. 29-41.

Lee, J. H. & Goldstein, R. J., 1988. An experimental study of natural

convection heat transfer in an inclined square enclosure containing internal

energy sources. ASME Journal of Heat Transfer, 110(2), pp. 345-349.

Liaqat, A. & Baytas, A. C., 2000. Heat transfer characteristics of internally

heated liquid pools at high Rayleigh numbers. Heat and Mass Transfer, 36(5),

pp. 401-405.

Liaqat, A. & Baytas, A. C., 2001. Conjugate natural convection in a square

enclosure containing volumetric sources. International Journal of Heat and

Mass transfer, 44(17), p. 3273–3280.

May, H. O., 1990. A numerical study on natural convection in an inclined

square enclosure containing internal heat sources. International Journal of

Heat and Mass Transfer, 34(8), pp. 919-928.

Misale, M., 2014. Overview on single-phase natural circulation loops.

Proceedings of the International Conference in Mechanical and Automatic

Engineering (MAE), Phuket, Thailand.

Misale, M. & Garibaldi, P., 2010. Dynamic behaviour of a rectangular singl-

phase natural circulation loop: influence of inclination.. ISHMT-ASME Heat

Transfer Conference, Mumbai, India.

Misale, M., Ruffino, P. & Frogheri, M., 2000. The influence of the wall

thermal capacity and axial conduction over a single-phase natural circulation

loop: 2-D numerical study. Heat and Mass Transfer, 36(6), pp. 533-539.

Ostrach, S., 1988. Natural convection in enclosures. ASME Journal of Haet

Transfer, 110(4b), pp. 1175-1190.

Pilkhwal, D. et al., 2007. Analysis of the unstable behaviour of a single-phase

natural circulation loop with one-dimensional and computational fluid-

dynamic models. Annals of Nuclear Energy, 34(5), pp. 339-355.

INTRODUCTION

9

Pini, A., Cammi, A. & Luzzi, L., 2016. Analytical and numerical investigation

of the heat exchange effect on the dynamic behaviour of natural circulation

with internally heated fluids. Chemical Engineering Science, Volume 145, pp.

108-125.

Pini, A., Cammi, A., Luzzi, L. & Ruiz, D. E., 2014. Linear and nonlinear

analysis of the dynamic behaviour of natural circulation with internally

heated fluid. Proceedings of 10th International Topical Meeting on Nuclear

Therma-Hydraulic, Okinawa, Japan.

Ruiz, D., Cammi, A. & Luzzi, L., 2015. Dynamic stability of natural

circulation loops for single phase fluids with internal heat generation.

Chemical Engineering Science, Volume 126, pp. 573-583.

Serp, J. et al., 2016. The Molten Salt Reactor (MSR) in generation IV:

Overview and perspective. Progress in Nuclear Energy, Volume 77, pp. 308-

319.

Shim, Y. M. & Hyun, J. M., 1997. Transient confined natural convection with

internal heat generation. International Journal of Heat and Fluid Flow, 18(3),

pp. 328-333.

Swapnalee, B. P. & Vijayan, P. K., 2011. A generalized flow equation for

single phase natural circulation loops obeying multiple friction laws.


Vijayan, P., 2002. Experimental observations on the general trends of the

steady state and stability behaviour of single-phase natural circulation loops.

Nuclear Engineering and Design, Volume 215, pp. 139-152.

Vijayan, P. K., Austregesilo, H. & Teschendorff, V., 1995. Simulation of the

unstable oscillatory behaviour of single phase natural circulation with

repetitive flow reversals in a rectangular loop using the computer code

ATHLET. Nuclear Engineering and Design, Volume 155, pp. 614-623.

Vijayan, P. K., Sharma, A. K. & Saha, D., 2007. Steady State and stability

characteristics of single-phase natural circulation in a rectangular loop with

different heater and cooler orientations. Experimental Thermal and Fluid

Science, Volume 31, pp. 925-945.

Welander, P., 1967. On the oscillatory instability of a differentially heated

fluid loop. Journal of Fluid Mechanics, 29(1), pp. 17-30.

Y. M. Shim, J. M. H., 1997. Transient confined natural convection with

internal heat generation. International Journal of Heat and Fluid Flow, pp.

328-333.

11

CHAPTER 1: Brief overview of Computational Fluid

Dynamics

1.1 Introduction

Computational Fluid Dynamics (CFD) is a branch of fluid mechanics whose

purpose is to solve the governing laws of fluid dynamics by means of

numerical simulations.

CFD codes are characterized by several advantages, such as the investigation

of flows in complex systems in a non-intrusive way and lower costs than

experiments. In spite of these advantages, it is important to underline that

CFD codes are not still able to totally replace experimental facilities, but

surely represent a useful complementary tool. Moreover, the accuracy of a

CFD code depends on the physical model adopted, on the errors induced by

the numerical discretization and on the initial (IC) and boundary (BC)

conditions considered.

Any CFD code is structured into three main steps: pre-processing, where the

system geometry is defined and the problem to be solved is set (definition of

fluid properties, physical model, IC and BC); solver, where the appropriate

numerical scheme is chosen for the discretization of the governing equations;

post-processing, for the data analysis.

Aim of this chapter is to introduce a physical background and the

mathematical instruments used hereafter in this work. The chapter is

organized as follows: in section 1.2, the governing equations of fluid dynamics

are briefly reported; section 1.3 gives an overview of turbulence modelling in

CFD codes; in section 1.4, some information about the Finite Volume Method

is reported; at last, the OpenFOAM® code is presented in section 1.5.

1.2 Navier-Stokes and energy equations

Governing equations of fluid dynamics are mathematically derived by

fundamental conservation laws of physics: conservation of mass, momentum

and energy.

In this work, the following assumptions are made on the working fluid:

The fluid is considered as incompressible with constant thermal

properties;

CHAPTER 1: Brief overview of Computational Fluid Dynamics

12

The Newtonian model is adopted for the link between shear stress and

momentum diffusivity;

Since we deal with natural convection phenomena, density gradients due

to variation of the temperature must be taken into account. In particular,

the Boussinesq approximation is considered;

Dissipative and diffusion terms in the energy equation are neglected;

Only uniform volumetric internal heat source is considered.

From the considerations reported above, the resulting set of PDEs to be

solved is summarized as follow:

∇ ∙ 𝒖 = 0 (1)

𝜕𝒖

𝜕𝑡+ ∇ ∙ (𝒖𝒖) = −

1

𝜌∗∇𝑝 + 𝜈∇2𝒖 +

𝜌(𝑇)

𝜌∗𝒈 (2)

𝜕𝑇

𝜕𝑡+ ∇ ∙ (𝑇𝒖) =

𝜈

𝑃𝑟∇ ∙ (∇𝑇) +

𝑞′′′

𝜌𝑐 (3)

𝜌(𝑇) = 𝜌∗[1 − 𝛽(𝑇 − 𝑇∗)] (4)

where Eq. (1) is the continuity equation, Eq. (2) represents the momentum

equation, Eq. (3) is the energy equation expressed in term of the temperature

of the fluid and Eq. (4) is the Boussinesq approximation, valid if 𝛽Δ𝑇 ≪ 1. All

the equations are reported in the conservative form.

In Eqs. (1) ÷ (4), 𝒖 is the velocity, 𝑝 is the pressure, 𝜈 is the momentum

diffusivity, 𝒈 is the gravitation acceleration, 𝜌 is the density, 𝛽 is the thermal

expansion coefficient, 𝑇 is the temperature field, 𝑃𝑟 is the Prandtl number of

the fluid, 𝑐 is the fluid specific heat and 𝑞′′′ is the volumetric internal energy

source. The superscript ∗ specifies the reference value for the thermo-physical

properties at the reference temperature 𝑇∗.

Solving Eqs. (1) ÷ (4) in case of natural convection with internal heat

generation involves several difficulties. Firstly, when buoyancy forces are

taken into account, the momentum and energy equations are coupled, forcing

to solve them simultaneously; secondly, the energy equation is no more a

homogeneous equation because of the presence of the internal energy source.

For these reason, Eqs. (1) ÷ (4), together with the appropriate auxiliary

conditions will be solved numerically.

1.3 Turbulence: theory and models

13


1.3.1 Characteristics of turbulence

In many applications of engineering interest the flow regime is turbulent and

for this reason it is necessary to study how the turbulence influences the

characteristics of the flow.

There is no univocal definition of turbulence. In general, when the inertial

forces of the fluid overcome the viscous forces, the flow starts to show a

random behaviour. In particular the flow switches from an ordered layer of

fluid structure, typical of laminar flow regime, to a chaotic rotational flows

structure, known as turbulent eddies, characterized by a wide range of length

scale.

Therefore, the main characteristics of turbulence flows are the random nature

and the turbulent eddies structure of the motion.

The generic flow variable 𝜙 can be expressed as the sum of a mean

component, which is independent of time when the mean flow is steady, and a

fluctuating component, which is always time-dependent, the so-called

Reynolds decomposition:

𝜙(𝑡) = Φ + 𝜙′(𝑡) (5)

where the operation of time-average is defined as:

�� =1

𝑇∫ 𝜙(𝑡)𝑑𝑡𝑇

0

≡ Φ (6)

and time-average of the fluctuating component is, by definition, zero:

1

𝑇∫ 𝜙′(𝑡)𝑇

0

≡ 0 (7)

Turbulent flow is intrinsically unsteady because of the chaotic component of

the motion. Nevertheless, turbulence is described by means of a deterministic

approach using the Navier-Stokes equations. Moreover, turbulent flow is

always 3D, even if the mean quantities have only a 1D or 2D spatial

dependence.

The turbulent eddies structure influences the characteristics of the flow in

two different ways: higher diffusivity and dissipation effects. The first regards

higher diffusion coefficients than laminar regime and derives from the mixing

effect of the eddies, improving mass, momentum and heat exchange. The


14

latter is a process known also as energy cascade and is linked to the several

length scale of the eddies. It consists in the energy transfer from larger eddies

to the smaller ones. In particular, the largest eddies extract kinetic energy

from the mean flow. Since viscous effects are in general negligible at this

length scale, the larger eddies give the smaller eddies almost all the energy

they received. At the smallest scale the viscous forces prevail and the energy

is converted into internal energy.

The energy spectrum can then be express as a function of wavenumbers (𝜅)

depending on the length scale of the eddies (ℓ):

𝐸 = 𝐸(𝜅), 𝜅 =2𝜋

ℓ (8)

An example of energy spectrum is shown in Figure 1.1.

The energy spectrum can be ideally divided into three different regions:

The first region is at low wavenumbers, typical of the largest eddies,

where the peak of the energy is found;

The third region is at high wavenumbers, typical of the smallest eddies,

where the energy reaches the lowest level due to the energy dissipation

process;

The second region is at intermediate wavenumbers. Kolmogorov derived

for this range of wavenumbers a spectral law in function of 𝜅 and the rate

of dissipation energy per unit mass 휀:

𝐸 ∝ 휀23⁄ 𝜅−

53⁄ (9)

Figure 1.1: Energy spectrum of turbulence (Versteeg & Malalasekera, 2007).


15

1.3.2 Turbulence models

For a turbulence model being useful in a CFD code, it must be applicable to a

wide range of turbulent flows, be economical to run ad as accurate and simple

as possible. Many efforts have been made in the development of numerical

methods for the study of turbulent flows based on these characteristics, at

different levels of detail. These methods can be grouped into three main

categories:

Reynolds-Averaged Navier-Stokes equations (RANS). It is obtained by

applying the Reynolds decomposition to the Navier-Stokes equations and

calculating the time-average. New terms arise in addition to the classical

ones, due to the correlation of the fluctuating components. This model

allows obtaining acceptable and useful information with reasonable

computational cost;

Large Eddy Simulation (LES). It takes into account different length scale

for the turbulent eddies by means of spatial filters. Large eddies are solved

numerically integrating the unsteady Navier-Stokes equations. The effects

of smaller eddies is included using sub-grid scale models, since they are

more homogenous and isotropic then larger eddies and therefore easier to

model. This method allows obtaining a more accurate solution increasing

the computational cost;

Direct Numerical Simulation (DNS). It is the most accurate method, and

solves the unsteady Navier-Stokes equations considering the effects of all

the fluctuations at any length scale. This method is rarely used for

industrial computations because of its high computational cost resulting

from the very fine spatial grids required.

In this work, different RANS turbulence models are used in order to compare

the results they provide. In the next sections, RANS equations are briefly

derived and a description of the Two-Equation models used is given.

1.3.3 RANS models

Consider an incompressible flow with constant viscosity. The classical Navier-

Stokes equations can be used to describe both laminar and turbulent flows

while the RANS equations allow investigating the effects of turbulent

quantities on the mean flow. These equations are obtained applying the

Reynolds decomposition to all the flow variables and time-averaging the

Navier-Stokes equations. For the generic flow variables 𝜙 and 𝜓 and the

generic coordinate 𝜉 we have:


16

�� = Φ,𝜕𝜙

𝜕𝜉

=𝜕Φ

𝜕𝜉, 𝜙𝜓 = ΦΨ+ 𝜙′𝜓′ (10)

so the equations become:

∇ ∙ 𝑼 = 0 (11)

𝜕𝑼

𝜕𝑡+ ∇ ∙ (𝑼𝑼) = −

1

𝜌∇𝑝 + 𝜈∇2𝑼− ∇ ∙ (𝒖′𝒖′ ) + 𝒈 (12)

𝜕𝑇

𝜕𝑡+ ∇ ∙ (𝑇𝑼) =

𝜈

𝑃𝑟∇2𝑇 −

1

𝜌∇ ∙ (𝜌𝑇′𝒖′ ) (13)

In the time-average process, new terms appear in the equations because of

the product of fluctuating components. These terms represent convective

transfer of flow variables due to turbulent eddies, as mentioned earlier. For

example, the extra terms in the momentum equations represent additional

stresses, known as Reynolds stresses, while the extra term in the energy

equation represent the effect of turbulence on heat transfer.

Purpose of RANS is to model these new quantities adding one or more

transport equations in order to close the system. Many of these models are

based on the so-called eddy viscosity assumption. It implies that an analogy

between viscous stresses and Reynolds stresses could exist. Newtonian model

of viscosity states that viscous stresses are proportional to the rate of

deformation of the fluid:

𝜏𝑖𝑗 = 𝜇 (𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖) , 𝑖, 𝑗 = 1,2,3 = 𝑥, 𝑦, 𝑧 (14)

Boussinesq proposed the same model for the Reynolds stresses, introducing a

turbulent viscosity 𝜇𝑡 and considering only the effect on the mean flow:

−𝑢𝑖′𝑢𝑗′ = 𝜈𝑡 (𝜕𝑈𝑖𝜕𝑥𝑗

+𝜕𝑈𝑗

𝜕𝑥𝑖) −

2

3𝑘𝛿𝑖𝑗 (15)

where 𝑘 is the turbulent kinetic energy per unit mass and it will be defined

later.

The last term represents a correction for the normal components of the

Reynolds stresses. However, an isotropic assumption is made on the Reynolds

stresses, even though this leads to inaccurate results. In fact, in RANS

turbulence models, the effects of the intrinsic anisotropy on mean flow is often

neglected and it is considered only in more complex non-linear two-equation

models and in the Reynolds Stress Model (seven-equation model).


17

The eddy viscosity assumption ca be applied also to all transport turbulent

terms, such as that in the energy equation, introducing a turbulent thermal

diffusivity 𝛼𝑡:

−𝜌𝑇′𝑢𝑖′ = 𝛼𝑡𝜕𝑇

𝜕𝑥𝑖 (16)

Since all these phenomena are due to the same mechanism, it is expected that

turbulent viscosity and turbulent thermal diffusivity have similar values

(Reynolds analogy). If we introduce a turbulent Prandtl number, defined as

the ratio between turbulent viscosity and turbulent diffusion coefficient of the

generic scalar, we can assume it has an almost constant value:

𝜎𝜙 =𝑣𝑡𝒟𝑡~𝑐𝑜𝑠𝑡 (17)

In this work, attention is paid to the classical two-equation turbulence

models, in particular: 𝑘 − 휀, 𝑘 − 𝜔 and 𝑆𝑆𝑇 𝑘 − 𝜔. All these models assume

that the Reynolds stresses are isotropic, which represents a limit in the

accuracy of the predictions for more complex flows. The transport quantities

considered are: the turbulent kinetic energy per unit mass 𝑘, the rate of

dissipation of turbulent kinetic energy per unit mass 휀 and the specific

dissipation 𝜔 = 𝑘 휀⁄ .

1.3.4 Equations for kinetic energy

The Reynolds decomposition can be applied also to the kinetic energy of the

fluid:

𝑘(𝑡) = 𝐾 + 𝑘 (18)

where:

𝐾 =1

2(𝑼 ∙ 𝑼), 𝑘 =

1

2(𝒖′ ∙ 𝒖′ ) (19)

Equations for mean flow kinetic energy and turbulent kinetic energy can be

derived taking the scalar product between the time-average Navier-Stokes

equations and the mean flow velocity and introducing the deformation tensor.

After cumbersome calculations we obtain:

𝜕𝐾

𝜕𝑡+ ∇ ∙ (𝐾𝑼) = ∇ ∙ (−

1

𝜌𝑃𝑼 + 2𝜈𝑼 ∙ 𝑆 − 𝑼 ∙ 𝒖′𝒖′ ) − 2𝜈𝑆 ∙ 𝑆 + 𝒖′𝒖′ ∙ 𝑆 (20)


18

𝑆𝑖𝑖 =𝜕𝑈𝑖𝜕𝑥𝑖

, 𝑆𝑖𝑗 =1

2(𝜕𝑈𝑖𝜕𝑥𝑗

+𝜕𝑈𝑗

𝜕𝑥𝑖) (21)

The introduction of the deformation tensor splits the effect of viscous stresses

and Reynolds stresses into two part: the second and the third terms on the

right-hand side of Eq. (20) represent the transport of the mean kinetic energy

due to viscous stresses and turbulence effects, respectively; the fourth and the

last terms on the right-hand side represent dissipation due to viscosity and

turbulence production, respectively.

The equation for the turbulent kinetic energy is derived in a similar manner,

taking the scalar product between the instantaneous Navier-Stokes equations

and the instantaneous velocity and subtracting Eq. (20) from the equation

just obtained:

𝜕𝑘

𝜕𝑡+ ∇ ∙ (𝑘𝑼) = ∇ ∙ (−

1

𝜌𝑝′𝒖′ + 2𝜈𝒖′ ∙ 𝑠′ −

1

2𝒖′ ∙ 𝒖′𝒖′ ) − 2𝜈𝑠′ ∙ 𝑠′ − 𝒖′𝒖′ ∙ 𝑆 (22)

𝑠𝑖𝑖′ =

𝜕𝑢𝑖′

𝜕𝑥𝑖, 𝑠𝑖𝑗

′ =1

2(𝜕𝑢𝑖

′

𝜕𝑥𝑗+𝜕𝑢𝑗

′

𝜕𝑥𝑖) (23)

Eq. (22) is similar to Eq. (20). Each term has the same physical meaning but

in Eq. (22) origin of change and transport of the turbulent kinetic energy is

due to the fluctuating components of the several flow quantities.

The fourth term on the right-hand side of Eq. (22) is always negative and

physically represents the energy transport from larger eddies to smaller. It is

known also as dissipation rate of turbulent kinetic energy per unit mass and

it can be modelled with the second equation of turbulence models:

휀 = 2𝜈𝑠′ ∙ 𝑠′ , [휀] = 𝑚2𝑠𝑒𝑐−3 (24)

Unfortunately, the exact 휀-equation cannot be used because of the presence of

many unknown and unmeasurable quantities.

The last term of the turbulent kinetic energy is equal in magnitude to that in

the mean kinetic energy but opposites in sign. It is possible to demonstrate

that in general the Reynolds stresses and the mean velocity gradient are

opposite in sign so this term is a production term for the turbulent kinetic

energy and a destruction term for the mean kinetic energy.

1.3.5 Standard 𝑘 − 휀 Model

In this model, the eddy momentum diffusivity is expressed by means of

velocity (𝒰) and length (ℓ) scale for large-scale turbulence, defined in function

of 𝑘 and 휀, as follow:


19

𝜈𝑡 ∝ 𝒰ℓ, 𝒰 = 𝑘12⁄ , ℓ =

𝑘32⁄

휀 ⇒ 𝜈𝑡 = 𝐶𝜇

𝑘2

휀 (25)

Even though 휀 is referred to smaller eddies scale, the rate of energy transfer

from mean flow to larger eddies is similar to the rate of energy dissipation of

smaller eddies if the flow is not characterized by abrupt changes, so the use of

휀 in defining the length scale is acceptable.

In the correct equation for 𝑘, some terms must be modified in order to

eliminate unknown variables. In particular, the rate of change, the transport

term by convection and the dissipation term are the same of the correct

equation while the transport term by diffusion and the production term

change.

As for the first, the term linked to the Reynolds stresses is modelled using the

eddy diffusivity model, introducing the turbulent Prandtl number for 𝑘:

1

2𝒖′ ∙ 𝒖′𝒖′ = −

𝑣𝑡𝜎𝑘∇𝑘 (26)

The corrective term in the Reynolds stresses is incorporated into the pressure

term and an effective pressure is defined:

𝑝𝑒𝑓𝑓 = 𝑝′ +

2

3𝜌𝑘𝛿𝑖𝑗 (27)

The introduction of this effective pressure doesn’t influence the results. In

fact this term is always neglected since it cannot be measured and its effects

are accounted for in the gradient diffusion term.

In the production term, the eddy momentum diffusivity assumption is used to

express the Reynold stresses:

−𝒖′𝒖′ ∙ 𝑆 = 2𝜈𝑡𝑆 ∙ 𝑆 −4

3𝑘∇ ∙ 𝑼 (28)

where the last term is equal to zero, from the continuity equation in case of

incompressible flows.

Additional terms must be taken into account if turbulence is generated also

by other forced (i.e. buoyancy force).

The final modelled 𝑘 equation for incompressible flow is:

𝜕𝑘

𝜕𝑡+ ∇ ∙ (𝑘𝑼) = ∇ ∙ [(𝜈 +

𝜈𝑡𝜎𝑘) ∇𝑘] − 휀 + 2𝜈𝑡𝑆 ∙ 𝑆 (29)


20

The equation for 휀 is obtained in a similar manner:

𝜕휀

𝜕𝑡+ ∇ ∙ (휀𝑼) = ∇ ∙ [(𝜈 +

𝜈𝑡𝜎𝜀) ∇휀] − 𝐶2𝜀

휀2

𝑘+ 𝐶1𝜀

휀

𝑘2𝜈𝑡𝑆 ∙ 𝑆 (30)

where 𝐶1𝜀 and 𝐶2𝜀 are dimensionless constants.

In spite of the simplicity and acceptable accuracy of the results, this model

presents several limits in its application.

The constants 𝐶𝜇, 𝜎𝑘, 𝜎𝜀, 𝐶1𝜀 and 𝐶2𝜀 are adjustable variables, chosen in

function of the turbulent flow. Unfortunately it has not been possible to find a

universal set of constants for all turbulent problems yet. General values

widely used are:

𝐶𝜇 = 0.09, 𝜎𝑘 = 1.0, 𝜎𝜀 = 1.30, 𝐶1𝜀 = 1.44, 𝐶2𝜀 = 1.92 (31)

The model works well with simple confined flows but moderate performance

are obtained with unconfined flows, linked to the incorrect modelling of the

production term in the 휀-equation. This problem can be overcome with ad hoc

adjustments of model constants 𝐶𝜀.

Another possible source of errors derives from neglecting the pressure term,

which leads to low accurate solutions for flows characterized by high pressure

gradients.

Last but not least is the impossibility to predict the effects on the mean flow

due to anisotropy of the Reynolds stresses, as mentioned earlier.

1.3.6 Wilcox 𝑘 − 𝜔 Model

In this model, the turbulent viscosity is expressed in function of a velocity

scale and a length scale but, unlike the 𝑘 − 휀 model, the length scale is

defined by means of the specific dissipation 𝜔:

𝜈𝑡 ∝ 𝒰ℓ, 𝒰 = 𝑘12⁄ , ℓ =

𝑘12⁄

𝜔 ⇒ 𝜈𝑡 =

𝑘

𝜔 (32)

The equations for 𝑘 and 𝜔 are derived in a similar way as seen in the previous

section:

𝜕𝑘

𝜕𝑡+ ∇ ∙ (𝑘𝑼) = ∇ ∙ [(𝜈 +

𝜈𝑡𝜎𝑘)∇𝑘] − 𝛽∗𝑘𝜔 + 2𝜈𝑡𝑆 ∙ 𝑆 (33)

𝜕𝜔

𝜕𝑡+ ∇ ∙ (𝜔𝑼) = ∇ ∙ [(𝜈 +

𝜈𝑡𝜎𝜔)∇𝜔] − 𝛽1𝜔

2 + 𝛾1𝜔

𝑘2𝜈𝑡𝑆 ∙ 𝑆 (34)


21

Even in this case, the effects on the mean flow due to anisotropy of the

Reynolds stresses are neglected. The values of the adjustable constants

proposed by Wilcox are:

𝜎𝑘 = 2.0, 𝜎𝜔 = 2.0, 𝛽∗ = 0.09, 𝛽1 = 0.075, 𝛾1 = 0.553 (35)

The main quality of this turbulence model is its more reliability in predicting

the near-wall behaviour than the 𝑘 − 휀 model but its greatest weakness is the

free-stream boundary condition. Turbulent kinetic energy, and consequently

turbulent specific dissipation, tends to zero and so the eddy viscosity is

indeterminate. This fact makes 𝑘 − 𝜔 model very sensitive to the choice of

conditions imposed for free-stream.

1.3.7 Menter Shear Stress Transport 𝑘 − 𝜔

This hybrid model is based on mixed characteristics of 𝑘 − 휀 and 𝑘 − 𝜔

models. The first is used for solving the free steam while the latter is used in

the near-wall region. This operation takes place with the transformation of

the 휀-equation into an 𝜔-equation, by substituting 휀 = 𝑘𝜔.

𝑘-equation remains unchanged while in the 𝜔-equation a new term appears

from the diffusion term in the 휀-equation during the transformation:

𝜕𝑘

𝜕𝑡+ ∇ ∙ (𝑘𝑼) = ∇ ∙ [(𝜈 +

𝑣𝑡𝜎𝑘)∇𝑘] − 𝛽∗𝑘𝜔 + 𝑃 (36)

𝜕𝜔

𝜕𝑡+ ∇ ∙ (𝜔𝑼) = ∇ ∙ [(𝜈 +

𝑣𝑡𝜎𝜔,1

)∇𝜔] − 𝛽2𝜔2 + 𝛾2

𝜔

𝑘2𝜈𝑡𝑆 ∙ 𝑆 +

2

𝜎𝜔,2𝜔∇𝑘 ∙ ∇𝜔 (37)

Several improvement was made for optimising the performance of the

𝑆𝑆𝑇 𝑘 − 𝜔:

The model constants are:

𝜎𝑘 = 1.0, 𝜎𝜔,1 = 2.0, 𝜎𝜔,2 = 1.17

𝛽∗ = 0.09, 𝛽2 = 0.083, 𝛾2 = 0.44 (38)

Limiters are used for turbulent kinetic energy and turbulent momentum

diffusivity in order to improve performance in stagnant flows and in flows

with adverse pressure gradient:

𝜈𝑡 =𝑎1𝑘

𝑚𝑎𝑥(𝑎1𝜔, 𝑆 𝐹2), 𝑃 = 𝑚𝑖𝑛 (2𝜈𝑡𝑆 ∙ 𝑆, 10𝛽

∗𝑘𝜔) (39)


22

where 𝑎1 is a constant, 𝑆 = (2𝑆 ∙ 𝑆)12⁄

and 𝐹2 is a blending function.

Use of blending functions is needed to avoid numerical instabilities. They can

arise in the computation of the turbulent momentum diffusivity and in the

transition between the two models. In particular, blending functions are used

for limiting the maximum value of 𝜈𝑡 and in the extra term in the 𝜔-equation.

1.3.8 Near-wall treatment

Near-wall region heavily influence the flow behaviour and represents one of

the major error source in turbulence models. An example is given by the 𝑘 − 휀

model where good estimations of flow quantities in the free-stream region can

be obtained but prediction in the near-wall region is its greatest weakness.

For this reason great attention has to be paid to this topic.

Numerical problems arise for standard 𝑘 − 휀 model when equations are

integrated in the near-wall region, because of the production term in the 휀-

equation. It is proportional to the inverse of turbulent kinetic energy but

𝑘 → 0 so 휀 → ∞. This problem doesn’t occur in the 𝑘 − 𝜔 model since the

production term in the 𝜔-equation doesn’t depend on the turbulent kinetic

energy and this explains why 𝑘 − 𝜔 model has better performance in the

near-wall region than 𝑘 − 휀 model. Wall functions have been developed in

order to avoid this kind of problems. 𝑘 − 휀 model is valid especially for fully

turbulent flows. High-Reynolds (HR) wall functions work well with values of

𝑦+ for which log-law is valid and relate local shear stresses with the mean

velocity, turbulent kinetic energy and dissipation rate energy. When 𝑦+~1,

corresponding to the viscous-sublayer, viscous stresses take over Reynolds-

stresses and viscous corrections are required. Low-Reynolds (LR) wall

functions have been developed to overcome this problem using Reynolds-

dependence wall-damping functions.

𝑘 − 𝜔 model can be used for both high and low turbulence without wall-

damping functions but it is sensitive to free-stream boundary conditions. This

is because turbulent viscosity is indeterminate for 𝜔 → 0.

𝑆𝑆𝑇 𝑘 − 𝜔 model presents positive aspects of both the preceding models. For

simple flows, 𝑆𝑆𝑇 𝑘 − 𝜔 model performances are similar to that of other

standard models. The strength of this model is linked to complex flows or

flows with adverse pressure gradient, where 𝑘 − 휀 and 𝑘 − 𝜔 models are less

accurate.

1.4 Finite volume method

23

1.4 Finite volume method

An important step in numerical computation is the discretization procedure.

Its purpose is to transform PDEs in a corresponding system of algebraic

equations. The physical domain must be firstly divided in a finite number of

discrete regions, named control volumes. Successively, the system of algebraic

equations previously obtained is solved in each cell for each time step.

Discretization can then be seen as a two-step procedure: discretization of the

physical domain; discretization of governing equation. As for the latter,

several methods have been developed: Finite Difference Method; Finite

Element Method; Finite Volume Method; Spectral Element Method.

The OpenFOAM® code is based on the Finite Volume Method (FVM), and for

this reason only this method is described.

Considering the general transport equation:

𝜕(𝜌𝜙)

𝜕𝑡+ 𝛻 ∙ (𝜌𝜙𝒖) = 𝛻 ∙ (𝒟𝛻𝜙) + 𝒮 (40)

where 𝜙 is the generic scalar property, Γ is its diffusion coefficient and 𝑆 the

source term. Integrating Eq. (41) in a control volume and in a small time-step

∆𝑡, and applying the Gauss’s divergence theorem:

∫𝜕

𝜕𝑡∫ 𝜌𝜙𝑑𝒱𝐶𝑉

𝑑𝑡Δ𝑡

+∫ ∫ 𝜌𝜙𝒖 ∙ 𝒏𝑑𝐴𝐴

𝑑𝑡Δ𝑡

=

= ∫ ∫ 𝒟𝛻𝜙𝒏𝑑𝐴𝐴

𝑑𝑡Δ𝑡

+∫ ∫ 𝒮𝑑𝒱𝐶𝑉

𝑑𝑡Δ𝑡

(41)

In the FVM, Eq. (41) must be satisfied in each control volume. The next step

is to discretize each term in Eq. (41). The general form of the discretized

equation at a fixed time-step is:

𝑎𝑃𝜙𝑃 =∑ 𝑎𝑛𝑏𝜙𝑛𝑏𝑛𝑏

+ 𝒮𝑢 (42)

where 𝑎 represent coefficients whose form depends on the discretization

scheme adopted, the subscripts 𝑃 and 𝑛𝑏 are referred to the point of the cell

considered and the neighbouring nodes adjacent to the cell itself, respectively,

and 𝒮𝑢 derives from the linearization of the source term (𝒮 = 𝒮𝑢 + 𝒮𝑃𝜙𝑃).

The final result is a linear system that can be represented in matrix

formulation as:


24

𝑀𝜙 = 𝑏 (43)

In general, the coefficient matrix 𝑀 is a square sparse matrix whose number

of non-zero elements depends on the discretization method adopted for each

term.

1.5 The OpenFOAM® code

1.5.1 OpenFOAM® structure

OpenFOAM® (Open Field Operation and Manipulation) (OpenFOAM, 2016) is

a free, open source software package, managed, maintained and distributed

by the OpenFOAM Foundation, based on C++ programming language. In this

work, we refer to OpenFOAM® 2.4.0.

The OpenFOAM® distribution contains a large amount of standard

applications divided into two main categories: solvers and utilities. These

applications allow for the solution of a wide range of scientific and

engineering problems, including magneto-hydro-dynamics, solid dynamics,

reaction kinetics, heat transfer and CFD, and also pre- and post-processing.

In addition to the various existing standard applications, one of the most

important features of OpenFOAM® is to offer users complete freedom in

creating or modifying solvers and utilities simply using the available

libraries. Other characteristics are the parallelization of almost all the

applications, which gives the opportunity to take full advantage of computer

hardware, and no license costs. A downside is represented, instead, by the

lack of a GUI (Graphical User Interphase).

As a consequence, all the setting are made by means of several files organized

in directories. In particular, as it can be noticed from any tutorial, each case

is organized into three main directories: 0; constant; system. In the

description of the structure of the code, we directly refer to the case of a CFD

problem. An example of the organization of the directories is reported in

Figure 1.2.

In the 0 directory, it is possible to set IC and BC for the flow quantities of

interest, including for example the pressure, velocity and temperature fields.

The constant directory contains files about mesh generation, in the sub-

directory polyMesh, and files about properties that could be considered fixed

for all the simulation, e.g. the gravitation acceleration, the physical properties

of the working fluid and, if it is needed, setting about turbulence model used.

1.5 The OpenFOAM® code

25

Figure 1.2: Example of directory structure.

At last, information on the solution procedure adopted for solving the problem

can be found in the system directory. In particular: in the controlDict file the

time simulation and the time-step are set, instructions on how to save data

output are reported and other useful setting, such as the Courant number, is

present; the choice of finite volume discretization schemes adopted for each

term of the governing equations (time discretization, diffusion terms,

divergence terms, laplacian terms) are specified in the fvSchemes file; in the

fvSolution file are reported linear equation solvers, tolerances and algorithm

controls. If the decomposeParDict file is present, it is possible to run the

simulation in parallel in several processors as many as specified in the file.

Depending on the physical problem to be solved, it is necessary to add other

files and directories, if required by the solver. Detail of the structure will be

presented for each case run in the following chapters.

1.5.2 OpenFOAM® solvers

Many solvers are already implemented in OpenFOAM® for treating heat

transfer and buoyancy-driven flows problems, from the laplacian equation of

the steady-state conductive heat transfer in solids to transient conjugate heat

transfer, where conduction in solid region and convection in an adjacent fluid

are solved simultaneously. A full list of heat transfer solvers can be found in

the official site of OpenFOAM®, in the OpenFOAM® User Guide or by typing

in the terminal “ls $FOAM_SOLVERS/heatTransfer”. In spite of the large

number of standard solvers, it could be necessary to implement specific

solvers, depending on the characteristic of physical problem to be solved.

In the particular case of this work, where natural convection heat transfer for

internally heated fluid must be studied, no solvers are present in


26

OpenFOAM®. For this reason, modified versions of standard solvers are

implemented.

The buoyantBoussinesqPimpleFoam is the standard solver used to solve non-

steady buoyancy-driven fluids with the Boussinesq approximation as link

between density and temperature fields, considering the fluid as

incompressible, and the PIMPLE algorithm for the pressure-velocity coupling.

However, it does not take into account multi-regions heat transfer. Starting

from this solver, the presence of an internal energy source is introduced

creating a new scalar field and modifying the energy equation file, TEqn.H,

by adding the last term in Eq. (3). The modified solver is named

buoyantSourceBoussinesqPimpleFoam.

Sometimes, the conductive heat transfer in a solid region can affect

significantly the temperature field of the adjacent flowing fluid. For this

reason, conjugate heat transfer must be also taken into account in order to

investigate this influence. The standard solver implemented to solve this kind

of problems in OpenFOAM® is the chtMultiRegionFoam. Its structure is quite

different from that of the previous solver, since it treats the fluid as

compressible. Even though this surely leads to more accurate results, the

computational cost is greater than the case of incompressible fluid with the

Boussinesq approximation. The introduction of the internal energy source is

made in a similar manner as previously done, modifying the fluid energy

equation, and the modified solver is named chtSourceMultiRegionFoam.

Further information on OpenFOAM® implementation will be reported for

each case in the following.

References

OpenFOAM, 2016, http://www.openfoam.com/; http://www.openfoam.org/.

OpenFOAM User Guide, http://www.openfoam.org/docs/.

Turbulence Models, http://www.cfd-online.com/Wiki/Two_equation_models.

Versteeg, H. & Malalasekera, W., 2007. An Introduction to Computational

Fluid Dynamics: The Finite Volume Method. s.l.:Pearson Education.

Wilcox, D., 1993b. Turbulence Modelling for CFD. La Canada, CS: DCW

Industries Inc.

http://www.openfoam.com/

http://www.openfoam.org/

http://www.openfoam.org/docs/

http://www.cfd-online.com/Wiki/Two_equation_models

29

CHAPTER 2: Natural circulation of fluids

characterized by an internal energy source in a

square cavity at different Rayleigh numbers

2.1 Introduction

Natural convection phenomena take place in many applications of interest

and for this reason they have been widely studied both experimentally and

numerically.

The development of more and more reliable CFD codes is based on their

validation with experimental data. One of the most common experimental

benchmark is referred to the buoyancy-driven cavity but, in spite of its

simplicity, a full description of these phenomena has not been obtained yet.

Moreover, even though many data are available in literature about low

Rayleigh number (𝑅𝑎) regimes, less studies have been conducted on transition

and turbulent regimes, because of their intrinsic unsteadiness and few

accuracy of the turbulence models adopted in numerical simulations.

Recently, more and more attention is paid to fluid with Internal Heat

Generation (IHG), especially for innovative engineering applications such as

that in nuclear power plants, whose main example is given by the Generation

IV MSR (GIF, 2014). The main effect of the presence of IHG is to affect the

dynamic behaviour of fluids in natural convection, leading to different flow

regimes which may induce hydrodynamic instabilities.

The aim of this chapter is to investigate the effect of IHG on a simple system

such that of the differentially heated square cavity. Firstly, the results

obtained with the OpenFOAM® code are compared with those of numerical

benchmarks. Successively, natural convection in the cavity is studied in a

wide range of external (𝑅𝑎𝑒𝑥𝑡) and internal (𝑅𝑎𝑖𝑛𝑡) Rayleigh numbers. In

particular, the external Rayleigh number varies in the range (106 ÷ 1.58 ×

109) and the internal Rayleigh number in the range (106 ÷ 1.58 × 1011),

considering several flow regimes. A comparison with the results obtained for

the square cavity in case of conventional natural circulation (i.e. without

IHG) is also reported in order to show the main differences induced by the

presence of an internal energy source. The effect of the Prandtl number of the

fluid is also examined, showing the differences between the results obtained

in OpenFOAM® and a similar numerical study at low Prandtl number in case

of only IHG.

CHAPTER 2: Natural circulation of fluids characterized by an internal energy source

in a square cavity at different Rayleigh numbers

30

In the second part of this chapter, Thermal Inertia (TI) of the cavity are taken

into account in order to introduce the treatment of multi-region systems.

When the system is characterized by imposed external temperatures, and in

absence of IHG, no effects of wall materials are expected on the dynamics of

the fluid inside the cavity, as confirmed by the results shown in the Appendix.

The presence of an internal energy source, in principle, can modify the results

with respect to the situation without TI. Consequently, for the sake of

completeness, the case with TI is also reported. Only the cases at 𝑅𝑎𝑒𝑥𝑡 = 106,

for the laminar flow regime, and 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109, for the turbulent flow

regime, are considered, and the cases 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100 and 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞

are examined for what concerns the effect of IHG. Two different materials are

chosen for the solid regions, namely AISI-316 and pure Aluminium, in order

to compare the results considering also different thermal properties for wall

materials.

2.2 Physical problem

The physical system consists of a 2D differentially heated square cavity with

IHG. The fluid in the cavity is considered as incompressible, with constant

thermal properties, and the Newtonian model is adopted for the link between

shear stress and momentum diffusivity. The problem is mathematically

described by the Navier-Stokes and energy equations reported in the previous

chapter.

Since we are interested in investigating the natural convection phenomena,

changes of the density due to temperature gradients must be taken into

account. These variations give rise to a buoyancy force which is the trigger

mechanism of the natural circulation. The Boussinesq approximation is

considered for relating density and temperature.

When an external temperature difference is applied and internal energy

source is present, two important dimensionless numbers must be introduced

in order to study the effects of external and internal heating separately. The

first is known as external Rayleigh number (𝑅𝑎𝑒𝑥𝑡) and corresponds to the

Rayleigh number used in conventional natural circulation, defined as in Eq.

(1):

𝑅𝑎𝑒𝑥𝑡 = 𝐺𝑟𝑒𝑥𝑡𝑃𝑟 =𝑔𝛽𝐿3Δ𝑇𝑃𝑟

𝜈2 (1)

where 𝐺𝑟 is the Grashof number 𝑔 is the gravitational acceleration, 𝛽 is the

thermal expansion coefficient, 𝐿 is the size of the cavity, Δ𝑇 denotes the

2.3 Modelling and implementation in OpenFOAM®

31

imposed external temperature difference, 𝑃𝑟 is the Prandtl number and 𝜈 the

momentum diffusivity.

The latter is the so-called internal Rayleigh number (𝑅𝑎𝑖𝑛𝑡) and represents

the effect of IHG natural convection phenomena. It is defined as:

𝑅𝑎𝑖𝑛𝑡 = 𝐺𝑟𝑖𝑛𝑡𝑃𝑟 =𝑔𝛽𝐿5𝑞′′′𝑃𝑟2

𝜈3𝜌𝑐𝑝 (2)

where 𝑞′′′ is the volumetric heat generation and 𝑐𝑝 is the fluid specific heat.

In this study, only uniform volumetric internal heat source is considered.

IC and BC are necessary to close the system. The fluid is initially at rest at a

uniform temperature 𝑇(𝑥, 𝑦, 0) = 𝑇𝑖𝑛𝑡. As regards the BC, the left side of the

cavity is at a fixed hot temperature 𝑇(0, 𝑦, 𝑡) = 𝑇ℎ, while the right side is at a

fixed cold temperature 𝑇(𝐿, 𝑦, 𝑡) = 𝑇𝑐. An adiabatic condition is ideally

imposed at the top and bottom walls. The no-slip condition is considered for

all the walls as BC for the velocity. The specific IC and BC considered are

reported in the following sections for each case.


Although the system is a 2D cavity, OpenFOAM® runs only with 3D

configurations, so a depth 𝐷 must be considered.

The code is run using the buoyantSourceBoussinesqPimpleFoam solver,

which consists in a modified version of the buoyantBoussinesqPimpleFoam

solver, already implemented in OpenFOAM®, where the effect of IHG is taken

into account simply by adding the volumetric internal heat source term in the

energy equation, as shown in the previous section. The PIMPLE algorithm is

used for the pressure-velocity coupling.

Meshes are generated with the blockMesh utility of the OpenFOAM® code. A

non-uniform grid (80 × 80) is adopted, finer in the near-wall region, where

more accurate solutions are required. Wall type is set for vertical and

horizontal walls of the cavity, where the BC are defined, while the empty type

is imposed to the walls in the third direction, so that a 2D geometry can be

considered.

Thermal properties values are reported in the transportConstant file. The

desired external Rayleigh number is obtained simply changing the value of

the gravitation acceleration, while the internal Rayleigh number is obtained

varying the value of the internal energy source, using the same value adopted

for the gravitation acceleration.



32

The flow regime is specified in the turbulenceProperties file: laminar is set for

laminar and transition flow regimes and RASModel is set for turbulent flow

regime. The particular turbulence model used is specified in the

RASProperties file.

The tolerance is set to 10−8 for all the physical variables, and the following

relaxation factors are imposed: 0.3 for pressure, 0.7 for velocity and 1 for

temperature. Bounded Gauss linear discretization is used as scheme for the

divergence terms while default discretization schemes are used for the other

terms. At last, the maximum Courant number is set equal to 1 in order to

obtain accurate solution.

2.4 Comparison between the OpenFOAM® code and

numerical benchmarks

Results are compared with the numerical solutions given by Shim & Hyun

(1997), where both the effect of external temperature difference and IHG are

considered. In Table 2.1, thermal IC and BC, the geometrical size of the

cavity and the values of thermal properties considered are summarized. A

simple sketch of the analysed system is shown in Figure 2.1.

The external Rayleigh number is set equal to 105, which corresponds to a

laminar flow regime, while two different values for the internal Rayleigh

number are considered, 106 and 107, in order to show the relative impact of

IHG on the flow pattern and temperature distribution. In the benchmark,

simulations start from the steady-state solution of the differentially heated

cavity with no IHG and the internal energy source is switched on at 𝑡 = 0. In

this work only the steady-state solution is considered and for this reason the

results are compared only with the last time step reported in the benchmark.

Table 2.1: Thermal IC and BC, size of the cavity, thermal properties at 300 K (Bergman et al., 2011)

𝑻𝒊𝒏 (𝑲) 𝑻𝒉 (𝑲) 𝑻𝒄 (𝑲) 𝑳 (𝒎) 𝑾 (𝒎) 𝜷 (𝑲−𝟏) 𝝂 (𝒎𝟐𝒔−𝟏) 𝑷𝒓

300 301.5 298.5 0.1 0.2 3 × 10−3 10−5 0.71

2.4 Comparison between the OpenFOAM® code and numerical benchmarks

33

Figure 2.1: Differentially heated 2D cavity with internal energy source.

Contour lines of stream function and temperature are shown for the two cases

and maximum dimensionless value for temperature and extreme

dimensionless values of the stream functions are reported. Dimensionless

temperature is defined as:

𝜃 =(𝑇 − 𝑇∗)

(𝑇ℎ − 𝑇𝑐) (8)

while the normalization factor for the stream function is not specified in the

article, so the comparison is made with the ratio between minimum and

maximum value.

Results are summarized in Table 2.2 with the Relative Errors (RE) and

contour lines are reported in Figures 2.2÷2.5. It can be noticed a good

agreement between solutions obtained with the OpenFOAM® code and the

data reported in Shim & Hyun (1997). The greatest RE is equal to ~2 %,

linked to the different numerical approach used in the simulation.

Table 2.2: Comparison with Benchmark

𝑹𝒂𝒊𝒏𝒕 𝑹𝒂𝒆𝒙𝒕⁄ = 𝟏𝟎 𝑹𝒂𝒊𝒏𝒕 𝑹𝒂𝒆𝒙𝒕⁄ = 𝟏𝟎𝟎

|𝝍𝒎𝒊𝒏 𝝍𝒎𝒂𝒙⁄ | 11.126 1.409

Shim & Hyun (1997) 10.897 1.401

RE 2.096 % 0.552 %

𝜽𝒎𝒂𝒙 0.865 5.49

Shim & Hyun (1997) 0.87 5.54

RE 0.575 % 0.903 %



34

Figure 2.2: Case 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡 = 10⁄ : Plots of stream function and isotherms (Shim & Hyun, 1997).

Figure 2.3: Case 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡 = 10⁄ : Contour lines obtained with OpenFOAM®.

Figure 2.4: Case 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡 = 100⁄ : Plots of stream function and isotherms (Shim & Hyun, 1997).

Figure 2.5: Case 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡 = 100⁄ : Contour lines obtained with OpenFOAM®.

2.5 Results

35

Symmetry breaking of the flow pattern is the first important effect of IHG.

When only an external temperature difference is applied, a single circulation

cell is present, symmetric with respect to the centre of the cavity. The

presence of IHG gives rise to an opposite buoyance force and a second

counter-circulation pattern is triggered. This fact is shown in Figure 2.3,

where the second pattern is located in the upper corner, near the hot wall.

Also isotherms are divided into two groups, finer near the cold wall. The

maximum of the temperature is greater than the temperature of the hot wall,

as reported in Table 2.2, and, as a consequence, heat is transferred from the

fluid to the environment.

All these effects are more evident when the ratio 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ is increased, and

so the effect of IHG becomes dominant. This time, the asymmetric counter-

circular pattern is of comparable size, isotherms are almost horizontal near

the top wall while temperature distribution is vertical near both hot and cold

walls. Maximum temperature is significantly larger than temperature of the

hot wall.

The effect of IHG is examined in detail in the following sections by varying

the ratio 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ .

2.5 Results

In this section, the same system considered previously for the assessment of

the solvers is studied in a wide range of the external and internal Rayleigh

number. In particular, four cases are considered: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case

(b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞, with

𝑅𝑎𝑒𝑥𝑡 = 106 ÷ 1.58 × 109. These cases represent only external heating,

comparable external-internal heating, dominant internal heating and only

IHG, respectively. Data on the case (a) are reported in Appendix.

Results are organized in several sections, in function of the external Rayleigh

number. In general, the quantities reported are:

Dimensionless 𝑢-velocity component at the mid-width with their locations;

Dimensionless 𝑣-velocity component at the mid-height with their

locations;

Maximum of the dimensionless temperature distribution;

Local Nusselt numbers (𝑁𝑢) at 𝑥 = 0 with the corresponding locations.

Maximum and minimum values for the velocity components are reported for

showing the symmetry breaking induced by the presence of the IHG in cases

(b) and (c), and maximum and minimum values and the value at mid-height

are reported for the Nusselt number. In case (d), only the maximum values of



36

the x-velocity component are reported at 𝑌 = 0.12, in order to show significant

values, and values of the Nusselt number are reported at three fixed location:

𝑌 = 0, 𝑌 = 0.5 and 𝑌 = 1. Normalization factors and further details will be

reported in each section.

For all the cases, some results are quoted with extra digits to better

appreciate small differences.

2.5.1 𝑅𝑎𝑒𝑥𝑡 = 106

For this value of the external Rayleigh number, laminar flow regime is

expected in case of no IHG. The dimensionless velocities, coordinates and

temperature are defined as follow:

𝑈 =𝑢𝐿

𝛼=𝑢𝐿𝑃𝑟

𝜈, 𝑉 =

𝑣𝐿

𝛼=𝑣𝐿𝑃𝑟

𝜈, 𝑋 =

𝑥

𝐿, 𝑌 =

𝑦

𝐿, 𝜃 =

𝑇 − 𝑇∗

𝑇ℎ − 𝑇𝑐 (9)

In case (d), where no external temperature difference is applied, the

dimensionless temperature is defined as:

𝜃 =𝑇

𝑇∗ (10)

The local Nusselt number is calculated using its definition, the Newton’s law

of cooling and the Fourier’s law applied to the fluid at the wall:

𝑁𝑢 =ℎ𝐿

𝜆, 𝑞′′ = ℎΔ𝑇, 𝑞′′ = −𝜆

𝜕𝑇

𝜕𝑥|𝑤𝑎𝑙𝑙

⇒ 𝑁𝑢 = −𝐿

Δ𝑇

𝜕𝑇

𝜕𝑥|𝑤

(11)

The factor 𝜕𝑇 𝜕𝑥⁄ |𝑤 is computed interpolating the first fifty points of the

numerical temperature distribution with a third grade polynomial and

calculating the first derivative for 𝑥 = 0.

In Tables 2.3÷2.5, the grid-independent study is reported for cases (b), (c) and

(d) with RE and a comparison between the four cases is summarized in Table

2.6. At last, comparison of velocity and thermal distributions with contour

lines of the stream functions and isotherms are shown in Figures 2.6 and 2.7.

2.5 Results

37

Table 2.3: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1

(𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 61.28961 61.30552 61.31027

RE / 0.026 % 0.008 %

𝒚𝒎𝒂𝒙 0.842 0.841 0.841

RE / 0.118 % 0

𝑼𝒎𝒊𝒏 -68.35078 -68.32671 -68.37463

RE / 0.035 % 0.070 %

𝒚𝒎𝒊𝒏 0.147 0.141 0.144

RE / 4.082 % 2.127 %

𝑽𝒎𝒂𝒙 212.23533 212.85232 213.1917

RE / 0.291 % 0.159 %

𝒙𝒎𝒂𝒙 0.036 0.036 0.036

RE / 0 0

𝑽𝒎𝒊𝒏 -224.07955 -224.81724 -225.22336

RE / 0.329 % 0.180 %

𝒙𝒎𝒊𝒏 0.964 0.964 0.964

RE / 0 0

𝜽𝒎𝒂𝒙 0.5 0.5 0.5

RE / 0 0

𝑵𝒖𝒎𝒂𝒙 17.0403 16.9467 16.885

RE / 0.550 % 0.364 %

𝒚𝒎𝒂𝒙 0.033 0.034 0.035

RE / 3.030 % 2.941 %

𝑵𝒖𝒎𝒊𝒏 0.635 0.6273 0.6353

RE / 1.207 % 1.275 %

𝒚𝒎𝒊𝒏 1 1 1

RE / 0 0

𝑵𝒖𝟏 𝟐⁄ 7.9423 7.9277 7.926

RE / 0.185 % 0.021 %



38


(𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 109.23918 109.45431 109.57359

RE / 0.197 % 0.109 %

𝒚𝒎𝒂𝒙 0.416 0.419 0.412

RE / 0.721 % 1.671 %

𝑼𝒎𝒊𝒏 -115.29903 -114.77008 -114.42928

RE / 0.459 % 0.297 %

𝒚𝒎𝒊𝒏 0.137 0.141 0.144

RE / 2.920 % 2.128 %

𝑽𝒎𝒂𝒙 90.39365 90.43909 90.4114

RE / 0.050 % 0.031 %

𝒙𝒎𝒂𝒙 0.842 0.85 0.848

RE / 0.950 % 0.235 %

𝑽𝒎𝒊𝒏 -431.53445 -432.29273 -436.30423

RE / 0.176 % 0.928 %

𝒙𝒎𝒊𝒏 0.974 0.972 0.976

RE / 0.205 % 0.411 %

𝜽𝒎𝒂𝒙 3.5617 3.5723 3.5783

RE / 0.299 % 0.168 %

𝑵𝒖𝒎𝒂𝒙 -100.8283 -101.3837 -101.326

RE / 0.550 % 0.057 %

𝒚𝒎𝒂𝒙 0.982 0.979 0.977

RE / 0.305 % 0.204 %

𝑵𝒖𝒎𝒊𝒏 -4.7957 -4.9107 -4.9983

RE / 2.398 % 1.785 %


RE / 0 0

𝑵𝒖𝟏 𝟐⁄ -28.6217 -28.902 -29.0747

RE / 0.979 % 0.597 %

2.5 Results

39

Table 2.5: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞

(𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 37.20656 37.26967 37.30312

RE / 0.170 % 0.090 %

𝒚𝒎𝒂𝒙 0.244 0.247 0.248

RE / 1.229 % 0.405 %

𝑽𝒎𝒂𝒙 35.61417 35.6136 35.6136

RE / 0.002 % 0.0008 %

𝒙𝒎𝒂𝒙 0.5 0.5 0.5

RE / 0 0

𝑽𝒎𝒊𝒏 -57.89773 -57.88254 -58.06060

RE / 0.026 % 0.307 %

𝒙𝒎𝒊𝒏 0.064 0.06 0.061

RE / 6.25 % 1.667 %

𝜽𝒎𝒂𝒙 1.0008 1.0008 1.0008

RE / 0 0

𝑵𝒖𝒀=𝟎 -0.1357 -0.1193 -0.1147

RE / 12.039 % 3.911 %

𝑵𝒖𝟏 𝟐⁄ -0.473 -0.484 -0.478

RE / 2.255 % 1.172 %

𝑵𝒖𝒀=𝟏 -0.62 -0.627 -0.603

RE / 1.183 % 1.172 %

Table 2.6: Comparison 𝑅𝑎 = 106

Case (a) Case (b) Case (c) Case (d)

𝑼𝒎𝒂𝒙 64.8941 61.3103 109.5736 37.3031

𝒚𝒎𝒂𝒙 0.85 0.841 0.412 0.248

𝑼𝒎𝒊𝒏 -64.8941 -68.3746 -114.4293 -37.3031

𝒚𝒎𝒊𝒏 0.15 0.144 0.144 0.752

𝑽𝒎𝒂𝒙 218.4251 213.1917 90.4114 35.6133

𝒙𝒎𝒂𝒙 0.04 0.036 0.848 0.5

𝑽𝒎𝒊𝒏 -218.4251 -225.2234 -436.3042 -57.8606

𝒙𝒎𝒊𝒏 0.96 0.964 0.976 0.061

𝜽𝒎𝒂𝒙 0.5 0.5 3.5783 1.0008

𝑵𝒖𝒎𝒂𝒙−(𝒀=𝟏) 17.6 16.885 -101.326 -0.603

𝒚𝒎𝒂𝒙 0.044 0.035 0.977 1

𝑵𝒖𝟏 𝟐⁄ 8.4 7.926 -29.0747 -0.478

𝑵𝒖𝒎𝒊𝒏−(𝒀=𝟎) 0.9 0.6353 -4.9983 -0.1147

𝒚𝒎𝒊𝒏 0.99 1 0 0



40


Figure 2.6: Comparison of the velocity distribution for 𝑅𝑎𝑒𝑥𝑡 = 10

6: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.


Figure 2.7: Comparison of the temperature distribution for 𝑅𝑎𝑒𝑥𝑡 = 10

6: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d)

𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.

From Tables 2.3÷2.5 it can be noticed that most of the RE reported are less

than 1 %. The only exceptions are represented by some of the locations of the

velocity components and the minimum of the Nusselt numbers. However, RE

remain below 4 %, which is an acceptable value if we consider that it is

referred to very low quantities.

Many considerations can be made from data reported in Table 2.6 and

Figures 2.6 and 2.7. First of all, the flow regime is laminar for all the cases, as

expected. In the cases (a) and (d), a perfect bilateral symmetric pattern can be

observed, as confirmed by data reported in Table 2.6. Case (b) is almost

identical to case (a), as shown from the magnitude of the velocity components

and their locations. All the considerations reported in Appendix are still valid

and the only effect of IHG is a slight asymmetry in the velocity profile but the

magnitude of the velocity components is of the same order of case (a). About

heat exchange, the maximum temperature is still equal to that of the hot

wall. Nusselt numbers slightly decrease because of the marginal effect of the

2.5 Results

41

opposite buoyancy force induced by IHG, but the direction of the heat flux

remains unchanged.

When the effect of IHG becomes dominant, as in case (c), symmetry breaking

takes place, as reported by Shim & Hyun (1997). In particular, near the hot

wall the effect of the opposite buoyance force due to IHG is appreciable and a

counter-circulation cell is developed, hindering the rising motion of the fluid.

In the other hand, near the cold wall the effect of IHG is superimposed on

that of the buoyance force due to external cooling, resulting in an increase of

the magnitude of y-velocity component, as shown in Table 2.6, where 𝑉𝑚𝑖𝑛 is

almost twice with respect to case (a). Horizontal thermal stratification is

clearly shown in Figure 2.7 (c). The maximum of the temperature is greater

than the temperature of the hot wall and this leads to an inversion in the

direction of the heat flux, as confirmed by the negative sign of the Nusselt

number. Moreover, heat exchange is greatly incremented outwards, as shown

by absolute Nusselt number values, up to eight times with respect to cases (a)

and (b).

The last case is that of only IHG. At this low internal Grashof number the

fluid moves along two symmetric counter-circulating cells. The magnitude of

the velocity components is less than the previous cases as well as the Nusselt

numbers, which remain below the unit. In fact the maximum of the

temperature slightly exceeds the temperature of the hot wall.

The presence of IHG at low internal Rayleigh number doesn’t affect the flow

regime. The only difference is that, when considering only IHG or dominant

effect of IHG, locations of the maximum and minimum of the Nusselt number

at the hot wall are inverted.

2.5.2 𝑅𝑎𝑒𝑥𝑡 = 107 ÷ 108

These values of external Rayleigh number are close to the limit value for

which transition from laminar to turbulent flow regime takes place in case of

no IHG, equal to ~2 × 108, as reported by Le Quéré (1991). A steady-state is

still reached, but the presence of IHG could affect the flow regime. The same

normalization factors of the previous section are used.

For 𝑅𝑎𝑒𝑥𝑡 = 107, the grid-independent study is reported for cases (b), (c) and

(d) in Tables 2.7÷2.9, with RE, and a comparison between the four case is

summarized in Table 2.10. Velocity and thermal distributions with contour

lines of the stream function and isotherms are shown in Figures 2.8 and 2.9.

Corresponding data for 𝑅𝑎𝑒𝑥𝑡 = 108 are summarized in Tables 2.11÷2.13 for

the grid-independent solutions and in Table 2.14 for the comparison. In

particular, only the minimum of the dimensionless y-velocity component and

the maximum of the dimensionless temperature are reported for the case at

𝑅𝑎𝑒𝑥𝑡 = 108 and 𝑅𝑎𝑖𝑛𝑡 = 1010 and a comparison with the results obtained with



42

𝑆𝑆𝑇 𝑘 − 𝜔 turbulent model is shown. The reason of this choice will be

explained in the following, while more information about turbulence model is

reported in the next section. Figures 2.10 and 2.11 show the velocity and

thermal distributions for all the cases.


(𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 139.82456 140.28819 140.50687

RE / 0.331 % 0.156 %

𝒚𝒎𝒂𝒙 0.885 0.885 0.878

RE / 0 0.791 %

𝑼𝒎𝒊𝒏 -154.5599 -155.24008 -155.53686

RE / 0.440 % 0.191 %

𝒚𝒎𝒊𝒏 0.125 0.124 0.122

RE / 0.8 % 1.613 %

𝑽𝒎𝒂𝒙 658.77137 678.44973 676.25796

RE / 2.987 % 0.323 %

𝒙𝒎𝒂𝒙 0.018 0.021 0.023

RE / 16.667 % 9.524 %

𝑽𝒎𝒊𝒏 -682.29012 -701.29895 -697.99035

RE / 2.786 % 0.472 %

𝒙𝒎𝒊𝒏 0.982 0.979 0.977

RE / 0.305 % 0.204 %

𝜽𝒎𝒂𝒙 0.5 0.5 0.5

RE / 0 0

𝑵𝒖𝒎𝒂𝒙 39.0463 39.3683 39.4003

RE / 0.825 % 0.081 %

𝒚𝒎𝒂𝒙 0.017 0.014 0.018

RE / 17.647 % 28.571 %

𝑵𝒖𝒎𝒊𝒏 0.929 0.9397 0.9263

RE / 1.148 % 1.419 %


RE / 0 0

𝑵𝒖𝟏 𝟐⁄ 14.522 14.533 14.522

RE / 0.076 % 0.007 %

2.5 Results

43


(𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 256.99018 252.83597 250.36375

RE / 1.616 % 0.978 %

𝒚𝒎𝒂𝒙 0.27 0.267 0.266

RE / 1.111 % 0.374 %

𝑼𝒎𝒊𝒏 -423.2949 -424.1547 -424.1739

RE / 0.203 % 0.004 %

𝒚𝒎𝒊𝒏 0.105 0.107 0.108

RE / 1.905 % 0.935 %

𝑽𝒎𝒂𝒙 141.44407 143.02382 143.79417

RE / 1.117 % 0.539 %

𝒙𝒎𝒂𝒙 0.874 0.876 0.871

RE / 0.229 % 0.571 %

𝑽𝒎𝒊𝒏 -1109.2614 -1131.7183 -1140.6789

RE / 2.024 % 0.792 %

𝒙𝒎𝒊𝒏 0.983 0.986 0.983

RE / 0.305 % 0.304 %

𝜽𝒎𝒂𝒙 2.2853 2.2997 2.3083

RE / 0.627 % 0.377 %

𝑵𝒖𝒎𝒂𝒙 -105.2323 -109.8037 -112.007

RE / 4.344 % 2.007 %

𝒚𝒎𝒂𝒙 0.991 0.986 0.988

RE / 0.505 % 0.203 %

𝑵𝒖𝒎𝒊𝒏 -3.274 -3.4707 -3.55

RE / 6.007 % 2.286 %


RE / 0 0

𝑵𝒖𝟏 𝟐⁄ -21.246 -21.3263 -22.102

RE / 0.378 % 3.636 %



44


(𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 84.24292 84.36788 84.43178

RE / 0.148 % 0.076 %

𝒚𝒎𝒂𝒙 0.218 0.216 0.215

RE / 0.917 % 0.463 %

𝑽𝒎𝒂𝒙 38.58807 38.50834 38.46425

RE / 0.207 % 0.114 %

𝒙𝒎𝒂𝒙 0.5 0.5 0.5

RE / 0 0

𝑽𝒎𝒊𝒏 -143.40935 -144.53399 -144.95076

RE / 0.784 % 0.288 %

𝒙𝒎𝒊𝒏 0.044 0.043 0.042

RE / 2.273 % 2.326 %

𝜽𝒎𝒂𝒙 1.0005 1.0005 1.0005

RE / 0 0

𝑵𝒖𝒀=𝟎 -0.091 -0.0963 -0.0867

RE / 5.861 % 10.035 %

𝑵𝒖𝟏 𝟐⁄ -0.426 -0.403 -0.433

RE / 5.40 % 7.527 %

𝑵𝒖𝒀=𝟏 -0.7523 -0.7477 -0.751

RE / 0.620 % 0.446 %



𝑼𝒎𝒂𝒙 148.7166 140.5069 250.3637 84.4318

𝒚𝒎𝒂𝒙 0.879 0.878 0.266 0.215

𝑼𝒎𝒊𝒏 -148.7166 -155.5369 -424.1739 -84.4318

𝒚𝒎𝒊𝒏 0.121 0.122 0.108 0.785

𝑽𝒎𝒂𝒙 695.7538 676.2580 143.7942 38.4642

𝒙𝒎𝒂𝒙 0.021 0.023 0.871 0.5

𝑽𝒎𝒊𝒏 695.7538 -697.9903 -1140.6789 -144.9508

𝒙𝒎𝒊𝒏 0.979 0.977 0.983 0.042

𝜽𝒎𝒂𝒙 0.5 0.5 2.3083 1.0005

𝑵𝒖𝒎𝒂𝒙−(𝒀=𝟏) 39.44 30.4 -112.007 -0.751

𝒚𝒎𝒂𝒙 0.018 0.018 0.988 1

𝑵𝒖𝟏 𝟐⁄ 14.91 14.522 -22.102 -0.433

𝑵𝒖𝒎𝒊𝒏−(𝒀=𝟎) 1.14 0.9263 -3.55 -0.0867

𝒚𝒎𝒊𝒏 0.99 1 0 0

2.5 Results

45






7: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ =0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.

This case gives substantially the same information reported for the case at

𝑅𝑎𝑒𝑥𝑡 = 106. When the effects of external and internal heating are

comparable, few differences are obtained with respect to case (a). Symmetry

starts to be broken but the difference between maximum and minimum

values of the velocities continues to be very low. Also the heat exchange is

practically unaffected by the IHG, as it can be noticed from Table 2.10, with

only a slight decrease of the Nusselt number.

In case (c) again symmetry breaking takes place in the same way as described

earlier and the heat flux is directed outwards, with values of the Nusselt

number higher than case (a) and (b).

Results obtained with only IHG present a regular and symmetric flow

pattern, as in the case of 𝑅𝑎𝑒𝑥𝑡 = 106. The maximum of the temperature

distribution is slightly less than the previous case, linked to a greater mixing

effect induced by the IHG. At this internal Rayleigh number, the

corresponding Grashof number is ~1.4 × 107 < 3 × 107, which represents the

expected value for symmetry breaking at low Prandtl number, as reported in

Arcidiacono et al. (2001).



46

Even in this case, no effects of IHG and of the Prandtl number on the flow

regime are found.


(𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟒𝟎 × 𝟏𝟒𝟎)

𝑼𝒎𝒂𝒙 279.38429 285.68412 288.7925

RE / 2.255 % 1.088 %

𝒚𝒎𝒂𝒙 0.918 0.92 0.92

RE / 0.218 % 0

𝑼𝒎𝒊𝒏 -339.44603 -349.35976 -354.13948

RE / 2.920 % 1.368 %

𝒚𝒎𝒊𝒏 0.066 0.067 0.063

RE / 1.151 % 5.970 %

𝑽𝒎𝒂𝒙 2076.5796 2129.9716 2118.1359

RE / 2.571 % 0.556 %

𝒙𝒎𝒂𝒙 0.014 0.012 0.01

RE / 14.286 % 16.667 %

𝑽𝒎𝒊𝒏 -2110.0064 -2169.2275 -2161.5737

RE / 2.807 % 0.353 %

𝒙𝒎𝒊𝒏 0.986 0.988 0.99

RE / 0.203 % 0.202 %

𝜽𝒎𝒂𝒙 0.5 0.5 0.5

RE / 0 0

𝑵𝒖𝒎𝒂𝒙 78.47 82.413 84.2847

RE / 5.029 % 2.271 %

𝒚𝒎𝒂𝒙 0.007 0.006 0.01

RE / 14.286 % 66.667 %

𝑵𝒖𝒎𝒊𝒏 1.3237 1.3297 1.3537

RE / 0.453 % 1.805 %


RE / 0 0

𝑵𝒖𝟏 𝟐⁄ 26.1093 26.1753 26.1907

RE / 0.253 % 0.059 %


Laminar Model (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

Laminar Model (𝟏𝟒𝟎 × 𝟏𝟒𝟎)

𝑺𝑺𝑻 𝒌 − 𝝎 Model (𝟑𝟏𝟎 × 𝟑𝟏𝟎)

𝑽𝒎𝒊𝒏 -3152.419 -3260.954 -3256.933

RE / 3.433 % 0.123 %

𝜽𝒎𝒂𝒙 1.4849 1.4908 1.4073

RE / 0.397 % 5.601 %

2.5 Results

47


(𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟒𝟎 × 𝟏𝟒𝟎)

𝑼𝒎𝒂𝒙 128.58526 129.28035 129.54873

RE / 0.540 % 0.208 %

𝒚𝒎𝒂𝒙 0.289 0.284 0.288

RE / 1.730 % 1.408 %

𝑽𝒎𝒂𝒙 85.25325 85.271 85.26532

RE / 0.021 % 0.007 %

𝒙𝒎𝒂𝒙 0.15 0.152 0.153

RE / 1.333 % 0.658 %

𝑽𝒎𝒊𝒏 -370.55326 -370.33103 -372.79757

RE / 0.060 % 0.666 %

𝒙𝒎𝒊𝒏 0.028 0.029 0.026

RE / 3.571 % 10.345 %

𝜽𝒎𝒂𝒙 1.0004 1.0004 1.0004

RE / 0 0

𝑵𝒖𝒀=𝟎 -0.0737 -0.0553 -0.07

RE / 24.887 % 26.506 %

𝑵𝒖𝟏 𝟐⁄ -0.3947 -0.3973 -0.3983

RE / 0.676 % 0.252 %

𝑵𝒖𝒀=𝟏 -0.916 -0.894 -0.9093

RE / 2.402 % 1.715 %



𝑼𝒎𝒂𝒙 330.9814 288.7925 400.2696 129.5487

𝒚𝒎𝒂𝒙 0.929 0.92 0.244 0.288

𝑼𝒎𝒊𝒏 -330.9814 -354.1395 -872.7462 -129.5487

𝒚𝒎𝒊𝒏 0.071 0.063 0.069 0.712

𝑽𝒎𝒂𝒙 2173.71 2118.1359 261.8842 85.2653

𝒙𝒎𝒂𝒙 0.012 0.001 0.059 0.153

𝑽𝒎𝒊𝒏 -2173.71 -2161.5737 -3256.933 -372.7976

𝒙𝒎𝒊𝒏 0.988 0.99 0.991 0.026

𝜽𝒎𝒂𝒙 0.5 0.5 1.4073 1.0004

𝑵𝒖𝒎𝒂𝒙−(𝒀=𝟏) 87.79 84.2847 -99.9147 -0.9093

𝒚𝒎𝒂𝒙 0.008 0.01 0,991 1

𝑵𝒖𝟏 𝟐⁄ 26.32 26.1907 -11.752 -0.3983

𝑵𝒖𝒎𝒊𝒏−(𝒀=𝟎) 2.3 1.3537 0 -0.07

𝒚𝒎𝒊𝒏 0.99 1 0.3538 0



48







At higher external Rayleigh number, the fluid starts to move along a more

and more narrow boundary layer, as shown in Figure 2.10(a). Velocity and

thermal fields sharply change in this region, so it is necessary to use finer

meshes. Greatest RE is those of locations, linked to both the boundary layer-

flow type and the considerations reported earlier, and the Nusselt number

near the corners, where flow sharply changes direction, giving less accurate

results.

For cases (a) and (b) nothing changes with respect to the previous cases, while

interesting information can be inferred from cases (c) and (d).

Case (c) deserves particular attention since it clearly shows the effect of IHG

on the flow regime.

As mentioned at the beginning of this section, when only the effect of an

external temperature difference is considered, laminar flow regime is

expected at 𝑅𝑎𝑒𝑥𝑡 = 108. This value is very close to the critical value predicted

by Le Quéré (1991) and for this reason the presence of an internal energy

source can play an important role in determining the flow regime. Running

2.5 Results

49

the simulation with the laminar model, a chaotic flow regime is obtained. It is

not possible to define if the flow regime is laminar or turbulent, since a

critical value of the Reynolds number (𝑅𝑒) for the cavity in presence of IHG is

not available in literature. For this reason both laminar and turbulent models

are considered in running the simulations and a comparison of the results is

reported in Table 2.12 as a first remark. Further studies will be necessary to

investigate how the transition from one flow regime to another occurs. This is

beyond the purpose of this work.

Data reported in the first two columns in Table 2.12 represent the mean

values for velocity and temperature, defined as:

�� ≡1

𝑇∫ 𝜙(𝑡)𝑑𝑡𝑇

0

(12)

where 𝜙 is the generic flow variable.

The integral is computed numerically in MATLAB® using the trapezoid rule:

��~∑[𝜙(𝑡𝑖+1) + 𝜙(𝑡𝑖)](𝑡𝑖+1 − 𝑡𝑖)

2𝑖 (13)

In Figures 2.12 and 2.13, the minimum of the dimensionless y-velocity

component and the maximum of the dimensionless temperature are reported

in function of time. It can be noticed that the amplitude of the oscillations is

still very small compared to the mean values calculated for both velocity and

temperature.

Figure 2.12: Plot Over Time and Mean Value of dimensionless v-velocity component (𝑅𝑎𝑒𝑥𝑡 = 10

8, 𝑅𝑎𝑖𝑛𝑡 = 1010).



50

Figure 2.13: Plot Over Time and Mean Value of dimensionless Temperature (𝑅𝑎𝑒𝑥𝑡 = 10

8, 𝑅𝑎𝑖𝑛𝑡 = 1010 ).

The greatest RE between grid (120 × 120) and grid (140 × 140) solutions

with laminar model is of the order of ~3 %, linked to the abrupt changes of

the velocity in the boundary layer region, and can be considered acceptable.

In the third column of Table 2.12 the same quantities obtained with 𝑆𝑆𝑇 𝑘 − 𝜔

model are compared with the grid-independent solution. The velocity is well

predicted while the maximum of temperature is underestimated in case of the

turbulence model because of the effect of the extra diffusion terms derived by

the eddy viscosity assumption. However, the RE with respect to the laminar

case is of 5.6 % and so the solution obtained with the turbulence model

represents a good approximation of the laminar case. The solution obtained

with 𝑆𝑆𝑇 𝑘 − 𝜔 model is considered in the comparison reported in Table 2.14.

Except the influence of the IHG on the flow regime, which represents one of

the most important features of fluids with internal energy source, most of the

considerations previously reported are still valid. As far as the heat exchange

is concerned, it can be noticed from the values of the Nusselt number and

from Figure 2.11(c) that the effect of IHG is less and less dominant for fixed

ratio 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ . In fact, in the cases at 𝑅𝑎𝑒𝑥𝑡 = 106 and 𝑅𝑎𝑒𝑥𝑡 = 107, the

Nusselt number at the hot wall is always negative, while in the case at

𝑅𝑎𝑒𝑥𝑡 = 108 the Nusselt number is both positive and negative, in the lower

and upper part respectively, with a minimum value of about ~0, that is no

heat exchange. This because the opposite buoyant force derived from the

presence of IHG is not strong enough to contrast the buoyancy force due to

the external temperature difference, at least in the lower region. The

inversion of the heat flux is approximately at 𝑌~0.35 ÷ 0.355.

All the other considerations about flow pattern and heat exchange discussed

earlier continue to be valid.

2.5 Results

51

2.5.3 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109

When only external heating was considered, a turbulent flow regime was

obtained (see Appendix). Simulations are run with a different setting with

respect to the previous cases. In particular, different size of the cavity and

thermal IC e BC are considered, taken from the experimental benchmark

given by Ampofo & Karayiannis (2003). These differences are summarized in

Table 2.15.

Following the experimental benchmark, a different definition is used for the

dimensionless velocity:

𝑉 =𝑣

√𝑔𝛽𝐿Δ𝑇 (14)

while the other dimensionless quantities are defined in the similar way of the

previous sections.

As far as the turbulence model adopted is concerned, RANS two-equations

models have been chosen. The choice is based on the results obtained in

Appendix for the conventional case. The best results are those obtained with

the standard 𝑘 − 휀 model with scalable wall functions and Menter 𝑆𝑆𝑇 𝑘 − 𝜔

without wall functions. In this study, only the Menter 𝑆𝑆𝑇 𝑘 − 𝜔 is used and

only a grid (310 × 310) is considered, due to the high computational cost

required from this kind of simulations.

Before comparing the results obtained in all the cases, it is of interest to

examine in detail the case of only IHG. In running the simulation, it was

observed that the dimensionless quantities obtained in case (d) appear to

depend on the size of the. This fact does not allow comparing the results with

those previously obtained (both with cases (d) at different Rayleigh numbers

and with the other cases at 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109). To solve this problem,

simulations are initially run with the same setting used in the previous

sections while the comparison with cases (a), (b) and (c) is made using the

setting reported in Table 2.15. As for the first set of simulations, grids

(120 × 120) and (140 × 140) are considered and simulations are run for

1000 𝑠𝑒𝑐 considering a laminar flow regime. A periodic flow regime is

obtained with an oscillation period equal to 360 𝑠𝑒𝑐.

Table 2.15: New Setting for the case at 𝑅𝑎 = 1.58 × 109


300 320 280 0.75 1.5 3.4 × 10−3 1.59 × 10−5 0.71



52

Figure 2.14: Plot Over Time and Mean Value of dimensionless v-velocity component (𝑅𝑎𝑖𝑛𝑡 = 1.58 × 10

9).

The transient behaviour is shown in Figure 2.14. In the same manner of case

(c) at 𝑅𝑎𝑒𝑥𝑡 = 108, the solutions obtained with the laminar model are

compared with that of the 𝑆𝑆𝑇 𝑘 − 𝜔 model. These results are summarized in

Table 2.16. This time the turbulence model does not give acceptable results if

compared with the laminar case, with a RE greater than 30 % for the y-

velocity component. This fact is evident since periodic flow is a regular flow

regime. Concerning the simulations run with the laminar model, the results

obtained with the grid (140 × 140) can be considered as grid-independent,

being the RE of the order of ~1 %.

From the results obtained in case of only IHG, it can be noticed that the

maximum of dimensionless temperature decrease as the internal Rayleigh

number increase. This fact is clearly shown in Eq. (15):

𝜃𝑚𝑎𝑥,6 = 1.0008 > 𝜃𝑚𝑎𝑥,7 = 1.0005 > 𝜃𝑚𝑎𝑥,8 = 1.0004 > 𝜃𝑚𝑎𝑥,9 = 1.0002 (15)

where the subscript is referred to the order of magnitude of the Rayleigh

number. This result is in agreement with that obtained by Arcidiacono et al.

(2001).

Table 2.16: Laminar vs. Turbulent flow regimes (Cavity ( 0.1 × 0.1))

Laminar Model (𝟏𝟐𝟎 × 𝟏𝟐𝟎)


𝑺𝑺𝑻 𝒌 − 𝝎 Model (𝟑𝟏𝟎 × 𝟑𝟏𝟎)

𝑽𝒎𝒊𝒏 -1121.347 -1138.843 -775.519

RE / 1.560 % 31.903 %

𝜽𝒎𝒂𝒙 1.00021 1.00021 1.000093

RE / 0 0.012 %

2.5 Results

53

If the same case (only IHG) is run using the setting in Table 2.15, a different

solution is obtained. This time the fluid manifests a chaotic flow regime, in

agreement with the prediction by Arcidiacono et al. (2001).

Also in this case, the results obtained in laminar flow regime are compared

with those of the 𝑆𝑆𝑇 𝑘 − 𝜔 model, as reported in Table 2.17. It can be noticed

a good agreement between the solutions is achieved, with a RE of ~1 %, and,

following the consideration reported for the case at 𝑅𝑎𝑒𝑥𝑡 = 108 and 𝑅𝑎𝑖𝑛𝑡 =

1010, the results obtained with the turbulence model will be adopted for the

comparison with cases (a), (b) and (c). This comparison is summarized in

Table 2.18. Velocity and thermal distributions with contour lines of the

stream function and isotherms are depicted in Figures 2.15 and 2.16.

Table 2.17: Laminar vs. Turbulent flow regimes (Cavity ( 0.75 ×0.75))


𝑺𝑺𝑻 𝒌 − 𝝎 Model (𝟑𝟏𝟎 × 𝟑𝟏𝟎)

𝑽𝒎𝒊𝒏 -0.03401 -0.03436

RE / 1.029 %

𝜽𝒎𝒂𝒙 1.00277 1.00261

R.E. / 0.016 %

Table 2.18: Comparison 𝑅𝑎 = 1.58 × 109( 𝑆𝑆𝑇 𝑘 − 𝜔, Grid (310 × 310))


𝑽𝒎𝒂𝒙 0.2421 0.2412 0.1283 0.00368

𝒙𝒎𝒂𝒙 0.007 0.007 0.007 0.907

𝑽𝒎𝒊𝒏 -0.2421 -0.2447 -0.3030 -0.03436

𝒙𝒎𝒊𝒏 0.993 0.993 0.995 0.984

𝜽𝒎𝒂𝒙 0.5 0.5 0.8366 1.0026

𝑵𝒖𝟏 𝟐⁄ 59.470 61.659 26.650 -0.3804


Figure 2.15: Comparison of the velocity distribution for 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 10




54


Figure 2.16: Comparison of the temperature distribution for 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 10


As observed in the previous section, an increase of the external Rayleigh

number lead to a less and less appreciable effects of the presence of the IHG.

It can be shown comparing the difference in magnitude of the y-velocity

components obtained in cases (b) with the corresponding value of the velocity

component in case of only external heating:

�� = ||𝑉𝑚𝑎𝑥(𝑏)| − |𝑉𝑚𝑖𝑛(𝑏)|

|𝑉(𝑎)|| (16)

Considering all the cases examined until now, we have:

��6 = 0.055 > ��7 = 0.031 > ��8 = 0.020 > ��9 = 0.014 (17)

where the subscript is referred to the order of magnitude of the Rayleigh

number.

This effect can also be observed in the flow pattern. In fact, the size of the

second counter-rotating pattern generated in case of dominant effect of IHG

(case (c)) decreases as the external Rayleigh number increase, as a

consequence of the increasing magnitude of the buoyancy force due to the

external imposed temperature difference.

As for the maximum of the temperature, it is less than the previous cases,

due to the increasing contribute of the convection at high external Rayleigh

number. The highness at which the inversion of the direction of the heat flux

is observed is greater than that reported for the case at 𝑅𝑎𝑒𝑥𝑡 = 108.

The local Nusselt number at the mid-height of the cavity is positive and no

inversion of the heat flux is observed at this point.

2.6 Effects of the Prandtl number on the flow regime

55


As well-known, fluids show different flow regimes. In case of natural

convection, the flow regime is determined by means of the Rayleigh number.

Most of the previous works in literature pay attention to fluids with a Prandtl

number of the order of one, and the flow regime is chosen setting the

appropriate value of the Grashof number. In Arcidiacono et al. (2001),

attention has been paid to the case of low-Prandtl number, showing different

flow regimes. The system always consists in square cavity, but only IHG is

taken into account. Several DNS simulations are run in a range of Grashof

number from 105 to 109 and the Prandtl number is set equal to 0.0321 (which

is representative of a lithium-lead alloy). Firstly, results are qualitatively

compared with those obtained with the OpenFOAM® code, keeping the same

setting previously used (see Table 2.1). In Figures 2.17÷2.20 contour lines of

stream function and isotherms are shown for four different cases.

At low Grashof number (𝐺𝑟 = 105), contribute of conduction and convection is

comparable and a steady-state is reached. Fluid flows along two perfectly

symmetric counter-circulation cells and vertical thermal stratification is

evident. Symmetry breaking of the flow pattern takes place at 𝐺𝑟 = 3 × 107.

Maximum of temperature is slightly smaller than the case of 𝐺𝑟 = 107, but

the temperature distribution is quite. As in the second case of Shim & Hyun

(1997), the maximum temperature is near the top wall, as a consequence of

the dominant effects of the natural circulation.

Figure 2.17: Case 𝐺𝑟 = 105.

Figure 2.18: Case 𝐺𝑟 = 3 × 107.



56

Figure 2.19: Case 𝐺𝑟 = 5.4 × 107.

Figure 2.20: Case 𝐺𝑟 = 108.

The effect of low Prandtl number is already shown in the third case. At

𝐺𝑟 = 5.4 × 107 a periodic flow regime is obtained. Both the symmetry breaking

and periodic oscillations was not predicted in the earlier work of Fusegi et al.

(1992), where a Prandtl number of 5.85 and an internal Rayleigh number of

5 × 109 are considered, which corresponds to 𝐺𝑟𝑖𝑛𝑡~8.5 × 108.

In the last case, in Arcidiacono et al. (2001) a chaotic flow regime is predicted

at 𝐺𝑟 = 108 and symmetry breaking is still observed. An increase of the

Grashof number to 109 leads to a fully chaotic regime. These differences

suggest that in case of IHG the flow regime not only depends on the internal

Grashof number, but it is affected by the Prandtl number too.

The results provided by Arcidiacono et al. (2001) are now compared with

those reported in the previous sections for the internal Rayleigh numbers

107 ÷ 1.58 × 109 and 𝑃𝑟 = 0.71. In Figures 2.21 and 2.22, comparisons of the

velocity and temperature distributions with contour lines of the stream

function and isotherms are shown for the three cases considered. In the first

case the corresponding internal Grashof number is equal to ~1.4 × 107 and a

laminar symmetric flow regime is obtained, coherently with the results at

low-Prandtl number. In this case the Prandtl number does not affect the flow

regime. In case of 𝑅𝑎𝑖𝑛𝑡 = 108, the effect of the Prandtl number can already be

observed. This time the internal Grashof number is ~1.4 × 108 and a

symmetric steady-state flow regime is obtained, in contrast with the case at

𝑃𝑟 = 0.0321, where a chaotic flow regime is predicted for 𝐺𝑟𝑖𝑛𝑡 ≥ 108.


57

(a) (b) (c)

(d) (e) (f)

Figure 2.21: Comparison of the velocity distribution: (a) 𝐺𝑟𝑖𝑛𝑡 = 10

7, (b) 𝐺𝑟𝑖𝑛𝑡 = 108,

(c) 𝐺𝑟𝑖𝑛𝑡 = 109, Arcidiacono et al. (2001); (d) 𝐺𝑟𝑖𝑛𝑡~1.4 × 10

7, (e) 𝐺𝑟𝑖𝑛𝑡~1.4 × 108, (f)

𝐺𝑟𝑖𝑛𝑡~2.2 × 109, OpenFOAM®.

(a) (b) (c)

(d) (e) (f)

Figure 2.22: Comparison of the temperature distribution: (a) 𝐺𝑟𝑖𝑛𝑡 = 10

7, (b) 𝐺𝑟𝑖𝑛𝑡 =108, (c) 𝐺𝑟𝑖𝑛𝑡 = 10

9, Arcidiacono et al. (2001); (d) 𝐺𝑟𝑖𝑛𝑡~1.4 × 107, (e) 𝐺𝑟𝑖𝑛𝑡~1.4 × 10

8, (f) 𝐺𝑟𝑖𝑛𝑡~2.2 × 10

9, OpenFOAM®.

Table 2.19: Effect of the Prandtl number on the flow regime (FR)

𝑷𝒓 = 𝟎. 𝟎𝟑𝟐𝟏 𝑷𝒓 = 𝟎. 𝟕𝟏 𝑮𝒓𝒊𝒏𝒕 = 𝟏𝟎

𝟓 ÷ 𝟑 × 𝟏𝟎𝟕 Symmetric FR Symmetric FR

𝑮𝒓𝒊𝒏𝒕 = 𝟑 × 𝟏𝟎𝟕 ÷ 𝟓. 𝟒 × 𝟏𝟎𝟕 Symmetry Breaking Symmetric FR

𝑮𝒓𝒊𝒏𝒕 = 𝟓. 𝟒 × 𝟏𝟎𝟕 ÷ 𝟏𝟎𝟖 Periodic FR Symmetric FR

𝑮𝒓𝒊𝒏𝒕~𝟏𝟎𝟖 Chaotic FR Symmetric FR



58

This fact clearly shows that, even for fixed internal Grashof number, the flow

regime can be affected by the Prandtl number of the fluid. Another evidence

is given by the case of 𝑅𝑎𝑖𝑛𝑡 = 1.58 × 109. As reported in section 2.5.3, a

periodic flow regime is observed (𝐺𝑟𝑖𝑛𝑡~2.2 × 109), in contrast with the case at

low Prandtl number, where a fully chaotic behaviour is predicted (𝐺𝑟𝑖𝑛𝑡 =

109). Asymmetric steady-state flow regime is not observed in the cases

considered for 𝑃𝑟 = 0.71 since it takes place in a very small range of Grashof

number.

The different flow regimes predicted in case of 𝑃𝑟 = 0.0321 and 𝑃𝑟 = 0.71 are

summarized in Table 2.19 for different range of the internal Grashof number.

2.7 Effects of thermal inertia

So far, the effect of TI of the cavity has been neglected. In the second part of

this chapter, two of the cases previously analysed are taken into account in

order to introduce the modified solver that will be used in the following of the

thesis for simulating the simultaneous presence of IHG and TI. In particular,

only the cases at 𝑅𝑎𝑒𝑥𝑡 = 106 and 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109 are considered for the

laminar and turbulent flow regime, respectively, and the cases (c)

(𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100) and (d) (𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞) are examined for what concern

the effect of IHG.

2.7.1 Implementation in OpenFOAM®

The dimensions of the cavity in the two cases are the same as those used in

the first part of this work, and reported in Table 2.1 and Table 2.15.

Following the same procedure adopted in Appendix, TI are introduced to the

left and right side with the same height and a width equal to an half of that of

the cavity. The simulations are run with the chtSourceMultiRegionFoam

solver.

Even in this case, meshes are generated with the blockMesh utility of the

OpenFOAM® code, and several grids are used in order to reach a grid-

independent solution. In particular, two uniform grids are considered for the

laminar flow regime, (100 × 100) and (120 × 120), and a non-uniform grid

(310 × 310) is used for the turbulent flow regime, taking as reference the

results obtained without TI, in the first part of this chapter. The

regionProperties file allows distinguishing solid and fluid regions, whose

spatial coordinates are defined in the topoSetDict file.

Thermal properties values of the air and of the solid walls are reported in the

transportProperties files placed in different dictionaries, which must be


59

created each time for different cases. In these files, it is necessary to define

the additional equation of state used to close the mathematical system,

transport properties and thermodynamics properties. As for the air

dictionary, it contains the file for defining the value of the gravitation

acceleration, which also in this case is modified in order to set the desired

value for the Rayleigh number, the turbulenceProperties file and the

RASProperties file. As far as the choice of turbulence model is concerned,

following the consideration reported in section 2.5.3, only Menter 𝑆𝑆𝑇 𝑘 − 𝜔

model without wall functions is considered. Table 2.20, summarizes the

relevant information for the air and for two different materials considered in

this study for the solid regions: AISI-316 and pure Aluminium.

The IC and BC for the cavity are almost the same as the previous cases, the

only differences being the BC at the left and right sides. This time, the fixed

hot and cold temperatures are imposed at the left side of the left solid region

and at the right side of the right solid region, respectively, and at the

interface solid-fluid the turbulentTemperatureCoupledBaffleMixed BC is

applied. Solid regions are initially at a uniform temperature equal to that

imposed as external BC. At last, the adiabatic condition is extended also to

other solid region boundaries.

As far as the numerical setting is concerned, even in this case a dictionary

must be created for each region. As for the air, the tolerance is set to 10−7 for

all the physical variables, and the same relaxation factors used earlier are

imposed. The upwind discretization is employed as scheme for the divergence

terms in both laminar and turbulent flow regimes, while the default

discretization schemes are used for the other terms. Default settings are used

for the solid regions too. At last, the maximum Courant number is set at 5

and 1 for laminar and turbulent flow regimes, respectively, in order to obtain

sufficiently accurate solution and to reduce the computational time.

Table 2.20: Thermo-Physical Properties (300 K)

Air AISI-316 Aluminium Eq. of state perfectGas (at 1 𝑎𝑡𝑚) rhoConst rhoConst

𝒄 (𝑱 𝒌𝒈−𝟏𝑲−𝟏) 1007 468 903

𝝆 (𝒌𝒈 𝒎−𝟑) 1.1614 8238 2702

𝝀 (𝑾 𝒎−𝟏𝑲−𝟏) - 13.4 237

𝝁 (𝑷𝒂 𝒔) 1.846 × 10−5 - -



60

2.7.2 Results

In this section, all the results obtained with the conjugate heat transfer solver

are summarized. In particular, the quantities reported are the same as those

considered in the previous sections. Simulations are conducted for 1000 𝑠𝑒𝑐

and 600 𝑠𝑒𝑐 for laminar and turbulent flow regimes, respectively.

In Tables 2.21 and 2.22 are reported the grid-independent studies for the

cases (c) and (d) at 𝑅𝑎𝑒𝑥𝑡 = 106, with the RE between two successive grids, for

all the materials considered for the solid regions, while in Tables 2.23 and

2.24 the same results are reported for the cases at 𝑅𝑎 = 1.58 × 109. A

comparison with the results obtained without thermal inertia is shown in

Tables 2.25 and 2.26. Figures of temperature and velocity distributions are

not reported since they are very similar to those reported in previous sections.

Table 2.21: Grid sensitivity for the TI case ( 𝑅𝑎𝑒𝑥𝑡 = 106, 𝑅𝑎𝑖𝑛𝑡 = 10

8)

AISI-316 Aluminium

(𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 101.6028 102.3265 102.7245 103.4173

RE / 0.712 % / 0.674 %

𝒚𝒎𝒂𝒙 0.41 0.408 0.41 0.408

RE / 0.488 % / 0.488 %

𝑼𝒎𝒊𝒏 -125.1458 -123.1267 -126.3701 -124.3373

RE / 1.613 % / 1.609 %

𝒚𝒎𝒊𝒏 0.1 0.1 0.1 0.108

RE / 0 / 8 %

𝑽𝒎𝒂𝒙 87.42081 87.53516 87.9023 88.0180

RE / 0.131 % / 0.132 %

𝒙𝒎𝒂𝒙 0.84 0.842 0.84 0.842

RE / 0.238 % / 0.238 %

𝑽𝒎𝒊𝒏 -410.8104 -422.1488 -414.2745 -425.6826

RE / 2.760 % / 2.754 %

𝒙𝒎𝒊𝒏 0.97 0.975 0.97 0.975

RE / 0.515 % / 0.515 %

𝜽𝒎𝒂𝒙 3.5003 3.509 3.4523 3.4607

RE / 2.248 % / 0.241 %

𝑵𝒖𝒎𝒂𝒙 -93.3247 -94.7733 -93.7733 -95.216

RE / 1.552 % / 1.538 %

𝒚𝒎𝒂𝒙 0.98 0.975 0.98 0.975

RE / 0.510 % / 0.510 %

𝑵𝒖𝒎𝒊𝒏 -5.203 -5.2 -5.1367 -5.1393

RE / 0.058 % / 0.052 %

𝒚𝒎𝒊𝒏 0 0 0 0

RE / 0 / 0

𝑵𝒖𝟏 𝟐⁄ -28.658 -28.7617 -28.2483 -28.3343

RE / 0.362 % / 0.304 %


61

Table 2.22: Grid sensitivity for the TI case ( 𝑅𝑎𝑒𝑥𝑡 = 0, 𝑅𝑎𝑖𝑛𝑡 = 106)

AISI-316 Aluminium

(𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼𝒎𝒂𝒙 34.9954 36.1172 35.0494 36.0448

RE / 3.205 % / 2.840 %

𝒚𝒎𝒂𝒙 0.25 0.25 0.25 0.25

RE / 0 / 0

𝑽𝒎𝒂𝒙 35.7471 35.8926 35.8760 35.9918

RE / 0.407 % / 0.323 %

𝒙𝒎𝒂𝒙 0.5 0.5 0.5 0.5

RE / 0 / 0

𝑽𝒎𝒊𝒏 -56.2913 -57.5876 -56.4172 -57.5349

RE / 2.303 % / 1.981 %

𝒙𝒎𝒊𝒏 0.06 0.059 0.06 0.059

RE / 1.667 % / 1.667 %

𝜽𝒎𝒂𝒙 1.0008 1.0008 1.0008 1.0008

RE / 0.0 % / 0.0 %

𝑵𝒖𝒀=𝟎 -0.1187 -0.119 -0.133 -0.1387

RE / 0.281 % / 4.261 %

𝑵𝒖𝒀=𝟏 -0.5653 -0.598 -0.566 -0.5997

RE / 5.778 % / 5.948 %

𝑵𝒖𝟏 𝟐⁄ -0.4577 -0.4797 -0.4513 -0.4793

RE / 4.807 % / 6.204 %

Table 2.23: TI results (𝑅𝑎𝑒𝑥𝑡 = 1.58 × 109, 𝑅𝑎𝑖𝑛𝑡 = 1.58 × 10

11)

AISI-316 Aluminium

𝑽𝒎𝒂𝒙 0.1140 0.1167

𝒙𝒎𝒂𝒙 0.007 0.007

𝑽𝒎𝒊𝒏 -0.2989 -0.3008

𝒙𝒎𝒊𝒏 0.995 0.995

𝜽𝒎𝒂𝒙 0.8791 0.8793

𝑵𝒖𝟏 𝟐⁄ 22.6927 23.2635

Following the same criteria used in the previous cases for the definition of

grid-independent solutions, from Table 2.21 it can be noticed that almost all

the RE are less than 1 % and the greatest errors are of the order of 2 % for

𝑉𝑚𝑖𝑛, which can be considered acceptable, while in Table 2.22 RE for the

Nusselt numbers reach value of 5-6 %, but referred to very small values.



62

Table 2.24: TI results ( 𝑅𝑎𝑒𝑥𝑡 = 0, 𝑅𝑎𝑖𝑛𝑡 = 1.58 × 109)

AISI-316 Aluminium

𝑼𝒎𝒂𝒙 0.02981 0.02962

𝒙𝒎𝒂𝒙 0.114 0.112

𝑼𝒎𝒊𝒏 -0.02992 -0.03005

𝒙𝒎𝒊𝒏 0.886 0.886

𝑽𝒎𝒂𝒙 0.00369 0.00359

𝒙𝒎𝒂𝒙 0.91 0.908

𝑽𝒎𝒊𝒏 -0.03474 -0.03485

𝒙𝒎𝒊𝒏 0.984 0.984

𝜽𝒎𝒂𝒙 1.0027 1.0027

𝑵𝒖𝒀=𝟎 -0.040 -0.049

𝑵𝒖𝒀=𝟏 -0.874 -0.881

𝑵𝒖𝟏 𝟐⁄ -0.398 -0.384

Table 2.25: Comparison with previous data (Laminar Flow Regime)

𝑹𝒂𝒆𝒙𝒕 = 𝟏𝟎

𝟔, 𝑹𝒂𝒊𝒏𝒕 = 𝟏𝟎𝟖 𝑹𝒂𝒆𝒙𝒕 = 𝟎, 𝑹𝒂𝒊𝒏𝒕 = 𝟏𝟎

𝟔 AISI-316 Aluminium No TI AISI-316 Aluminium No TI

𝑼𝒎𝒂𝒙 102.3265 103.4173 109.5736 36.1172 36.0448 37.3031

RE 6.614 % 5.618 % / 3.179 % 3.373 % /

𝒚𝒎𝒂𝒙 0.408 0.408 0.412 0.25 0.25 0.248

RE 0.971 % 0.971 % / 0.806 % 0.806 % /

𝑼𝒎𝒊𝒏 -123.1267 -124.3373 -114.4293 -36.1172 -36.0448 -37.3031

RE 7.601 % 8.659 % / 3.179 % 3.373 % /

𝒚𝒎𝒊𝒏 0.1 0.108 0.144 0.75 0.75 0.752

RE 30.556 % 25 % / 0.267 % 0.267 % /

𝑽𝒎𝒂𝒙 87.5352 88.0180 90.4114 35.8926 35.9918 35.6133

RE 3.181 % 2.647 % / 0.784 % 1.063 % /

𝒙𝒎𝒂𝒙 0.842 0.842 0.848 0.5 0.5 0.5

RE 0.708 % 0.708 % / 0 0 /

𝑽𝒎𝒊𝒏 -422.1488 -425.6826 -436.3042 -57.5876 -57.5349 -57.8606

RE 3.244 % 2.434 % / 0.472 % 0.563 % /

𝒙𝒎𝒊𝒏 0.975 0.975 0.976 0.059 0.059 0.061

RE 0.102 % 0.102 % / 3.279 % 3.279 % /

𝜽𝒎𝒂𝒙 3.5003 3.509 3.5782 1.0008 1.0008 1.0008

RE 2.177 % 1.934% / 0 0 /

𝑵𝒖𝒎𝒂𝒙 -94.7733 -95.216 -101.326 -0.598 -0.5997 -0.603

RE 6.467 % 6.036 % / 0.829 % 0.547 % /

𝒚𝒎𝒂𝒙 0.975 0.975 0.977 1 1 1

RE 0.205 % 0.205 % / 0 0 /

𝑵𝒖𝟏 𝟐⁄ -28.7617 -28.3343 -29.0747 -0.4797 -0.4793 -0.478

RE 1.077 % 1.547 % / 0.356 % 0.272 % /

𝑵𝒖𝒎𝒊𝒏 -5.2 -5.1393 -4.9983 -0.119 -0.1387 -0.1147

RE 4.035 % 2.821 % / 3.749 % 20.924 % /

𝒚𝒎𝒊𝒏 0 0 0 0 0 0

RE 0 0 / 0 0 /

2.8 Final remarks

63

Table 2.26: Comparison with previous data (Turbulent Flow Regime)

𝑹𝒂𝒆𝒙𝒕 = 𝟏. 𝟓𝟖 × 𝟏𝟎

𝟗, 𝑹𝒂𝒊𝒏𝒕 = 𝟏. 𝟓𝟖 × 𝟏𝟎𝟏𝟏 𝑹𝒂𝒆𝒙𝒕 = 𝟎, 𝑹𝒂𝒊𝒏𝒕 = 𝟏. 𝟓𝟖 × 𝟏𝟎

𝟗 AISI-316 Aluminium No TI AISI-316 Aluminium No TI

𝑽𝒎𝒂𝒙 0.1140 0.1167 0.1284 0.00369 0.00359 0.00368

RE 11.215 % 9.112 % / 0.272 % 2.446 % /

𝒙𝒎𝒂𝒙 0.007 0.007 0.007 0.91 0.908 0.907

RE 0 0 / 0.331 % 0.110 % /

𝑽𝒎𝒊𝒏 -0.2989 -0.3008 -0.3030 -0.03474 -0.03485 -0.03438

RE 1.353 0.726 % / 1.047 % 1.367 % /

𝒙𝒎𝒊𝒏 0.995 0.995 0.995 0.984 0.984 0.984

RE 0 0 / 0 0 /

𝜽𝒎𝒂𝒙 0.8791 0.8793 0.8366 1.0027 1.0027 1.0026

RE 5.080 % 5.104 % / 0.001 % 0.001 % /

𝑵𝒖𝟏 𝟐⁄ 22.6927 23.2635 26.6503 -0.398 -0.384 -0.380

RE 14.850 % 12.083 % / 4.737 % 1.053 % /

As for the comparison with/without TI, for both the laminar and turbulent

flow regimes, RE from 2 % to 5 % are obtained for the maximum of the

dimensionless temperature, while RE from 4 % to 12 % for the Nusselt

number. RE for the velocity components vary from 6 % to 10 %. Considering

the RE values, no differences between the cases with/without TI can be

highlighted, at least within the level of detail of the adopted model.

2.8 Final remarks

In this chapter, numerical studies are adopted in order to assess the modified

solvers of OpenFOAM® code in case of system in natural circulation with

fluids heated both externally and internally. Good agreement is obtained

between simulations and the numerical benchmarks provided by Shim &

Hyun (1997) and, qualitatively, with Arcidiacono et al. (2001).

The main purpose is to study how the presence of internal energy source can

affect the dynamics in natural convection. Different flow regimes are

considered, by varying the external Rayleigh number in the range 106 ÷

1.58 × 109, and four different cases are considered: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0;

case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.

The general flow pattern in the case of classical buoyant cavity is that in

which the fluid moves long a more and more narrow boundary layer, where

velocity and temperature change sharply, whereas in the core region of the

cavity the air is overall at rest.

The main effects of the presence of IHG are: the change in the flow pattern,

as already shown in case (c) for all the external Rayleigh numbers,

characterized by the symmetry breaking and a second counter-rotating cell

near the hot wall, generated by an opposite buoyance force due to the



64

presence of IHG; the influence on the flow regime, as observed in the case of

𝑅𝑎𝑒𝑥𝑡 = 108 and 𝑅𝑎𝑖𝑛𝑡 = 1010, where it can be noticed that the presence of an

internally heated fluid and an external Rayleigh number very close to the

critical value lead to a change in the flow regime, from laminar to chaotic.

Another interesting effect is the influence of the Prandtl number of the

working fluid on the flow regime, at fixed Rayleigh number. Comparing the

results obtained using air as fluid (𝑃𝑟 = 0.71) with those given by Arcidiacono

et al. (2001) (𝑃𝑟 = 0.0321) in case of only IHG, differences are already

obtained at 𝑅𝑎𝑖𝑛𝑡 = 107. This effect underlines how the choice of the working

fluid itself contributes in determining the flow regime and so the dynamic

behaviour in natural circulation.

As far as the turbulence model is concerned, following the results obtained in

Appendix, the 𝑆𝑆𝑇 𝑘 − 𝜔 model without wall function is used and good results

are obtained also in case of IHG. At this step, it is not possible to define if the

flow regime is laminar or turbulent, since data on this topic are not available

in literature, at least for a differentially heated cavity with internal energy

source. The results obtained with the turbulence model, however, are in good

agreement with those obtained with a laminar model.

At last, thermal inertia are considered with the primary aim to introduce the

conjugate heat transfer modified solver used in chapter 4. For the considered

system, the presence of thermal inertia does not influence the solutions

obtained, at least within the level of the detail of the adopted model.

References


turbulent natural convection in an air filled square cavity. Internation


Arcidiacono, S., Di Piazza, I. & Ciofalo, M., 2001. Low-Prandtl number

natural convection in volumetrically heated rectangular anclosures II. Square

cavity, AR=1. International Journal of Heat Transfer, Issue 44, pp. 537-550.

Bergman, T. L., Lavine, A. S., Incropera, F. P., DeWitt, D. P., 2011.

Fundamentals of Heat and Mass Transfer. John Wiley & Sons, Inc., Hoboken,

NJ, United States.

Fusegi, T., Hyun, J. M. & Kuwahara, K., 1992. Numerical study of natural

convection in a differentially heated cavity with internal heat generation.

ASME International Journal of Heat Transfer, Issue 144, pp. 773-777.

References

65



Shim, Y. M. & Hyun, J. M., 1997. Transient confined natural convection with

internal heat generation. International Journal of Heat and Fluid Flow, 18(3),

pp. 328-333.




Industries Inc.

67

CHAPTER 3: Dynamic stability for single-phase

natural circulation in rectangular loops

3.1 Introduction

Natural circulation systems are usually rectangular or toroidal loops, in

which the working fluid transfers heat between a hot source and a cold sink

exploiting the driving action of the buoyancy force.

In literature, the dynamic behaviour of natural circulation loops have been

studied both theoretically and experimentally in several works. The first

theoretical studies were carried out by Keller (1966) and Welander (1967),

and more recently by Chen (1985), Vijayan et al. (1995), Misale et al. (2000,

2010), Pilkhwal et al. (2007), and Swapnalee & Vijayan (2011). Experimental

studies were performed by Vijayan (2002, 2007) and Swapnalee & Vijayan

(2011).

According to Misale (2014), natural circulation dynamics can be classified as

either stable or unstable depending on its time development. The equilibrium

state is achieved when the driving buoyancy force is in balance with the

frictional one. If the fluid motion is stable, the velocity and the temperature

distributions reach a steady-state value. In case of unstable conditions, the

fluid flow is characterised by oscillations of both the velocity and the

temperature. This oscillating behaviour can be unidirectional (i.e., the main

flow direction does not change) or can induce flow reversals.

In this chapter, a 3D CFD study on a single-phase natural circulation loop is

conduct by means of the OpenFOAM® (OpenFOAM®, 2016) code, comparing

the results obtained with the experimental data provided by the L2 facility of

DIME-TEC Labs (Genova University) and reported by IAEA (2014). The

rectangular loop is studied at different levels of detail, more and more close to

the experimental rig, in order to find the setting which better reproduces the

experimental results provided by the benchmark. In particular, three

configurations are considered, namely: only the working fluid; conjugate heat

transfer between the working fluid and pipe wall materials; a complete

coaxial heat exchanger model for the heat sink.

This chapter is organized as follows. In section 4.2, the description of L2

testing facility is presented. Section 4.3 deals with the techniques used for the

semi-analytical stability analysis. In section 4.4, the numerical models

adopted are presented, and in section 4.5 the comparison of the results with

CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular

loops

68

the experimental data is reported. In section 4.6, the main conclusions are

drawn.

3.2 Experimental set-up and procedure

The experimental data used in this work are provided by the L2 facility

installed at DIME-TEC Labs of Genova University. It consists on a vertical

rectangular loop with a circular cross-section of constant diameter, composed

of a single heat source and a single heat sink, called heater and cooler,

respectively. The former is placed lower than the latter in order to ensure the

development of natural circulation. In particular, the heater is placed in the

bottom horizontal leg of the loop while the cooler in the top one, in a

Horizontal-Heater Horizontal-Cooler (HHHC) configuration. A picture of the

experimental rig is shown in Figure 3.1 while the facility scheme is reported

in Figure 3.2. Geometric data of the loop are summarized in Table 3.1. 𝐷, 𝑊

and 𝐻 are the internal diameter, the width and the height of the rectangular

loop, respectively; 𝐿ℎ and 𝐿𝑐 represent the length of the heater and of the

cooler, respectively; 𝐿𝑡𝑜𝑡 is the total length and 𝐻 𝑊⁄ is the aspect ratio.

The heating and the cooling sections are made of pure Copper (99.9 %) while

pipes of several materials can be employed for the vertical branches and the

four bends. The insulation of the vertical pipes is realized thanks to an

Armaflex® layer (𝜆 = 0.038 𝑊 𝑚−1𝐾−1 at 40 ℃), ensuring a quasi-adiabatic

condition.

Figure 3.1: L2 experimental rig picture (IAEA, 2014).

Figure 3.2: L2 experimental facility

scheme.

3.2 Experimental set-up and procedure

69

Table 3.1: L2 experimental rig dimensions

𝑫 (𝒎𝒎) 𝑾 (𝒎𝒎) 𝑯 (𝒎𝒎) 𝑳𝒉 (𝒎𝒎) 𝑳𝒄 (𝒎𝒎) 𝑳𝒕𝒐𝒕 (𝒎𝒎) 𝑳𝒕𝒐𝒕 𝑫⁄ 𝑯 𝑾⁄

30 1112 988 960 900 4100 136.67 0.88

A uniform heat flux is provided electrically by means of a Nicromel wire

wrapped uniformly around the heating section and connected to a

programmable DC power supply. The cooling section is realized with a coaxial

tube connected to a cryostat in order to keep fixed the temperature of the

coolant, which consists of a mixture water-glycol. The temperature difference

of the coolant is maintained below 1 𝐾 thanks to a suitable flow rate imposed

by an external pump.

An expansion tank is present in order to allow working fluid to expand,

maintaining the internal pressure equal to the atmospheric one. Accurate

measures of the temperature of both the solid and the fluid regions are

provided by the several calibrated thermocouples placed in different sections

of the loop, some of which are depicted in Figure 3.3.

During the several experimental campaigns conducted at the DIME-TEC

Labs, the effect of different parameters was investigated, including: piping

materials; geometry and inclination of the loop; the choice of the working

fluid; the thermal power at the heating section; the heat sink temperature. In

this work reference is made to the data available from IAEA (2014).

The loop is placed in vertical position, water is used as working fluid and

stainless steel AISI-304 is adopted for vertical pipes and bends. Each run

starts from stagnant condition of the fluid, whose can be experimentally

verified from a negligible difference in the temperatures measured from the

several thermocouples. A fixed thermal power of 2000 𝑊 is provided at the

heater and different heat sink temperature are considered, from 4 ℃ to 18 ℃,

in order to show how this choice affects the dynamic behaviour of the system.

Figure 3.3: Sketch of the position of the main thermocouples (Misale & Garibaldi, 2010).


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70

Table 3.2: Experimental campaign test matrix

Experiment index

Heat Power (𝑾)

Heat sink temperature (℃)

Number of runs

Runs index Dynamic

behaviour

1 2000 4 3 (a, b, c) Unstable 2 2000 5 1 - Unstable 3 2000 7 1 - Stable

4 2000 8 5 (a, c, d) Stable

(b, e) Unstable 5 2000 9 1 - Stable 6 2000 10 2 (a, b) Stable 7 2000 11 2 (a, b) Stable 8 2000 14 2 (a, b) Stable 9 2000 18 1 - Stable

The experimental conditions are reported in Table 3.2. It can be noticed an

unstable dynamics characterized by bi-directional oscillation at low heat sink

temperature (4 ℃ and 5 ℃) while a stable equilibrium is reached at higher

temperature. It can be noticed an unstable dynamics characterized by bi-

directional oscillation at low heat sink temperature (4 ℃ and 5 ℃) while a

stable equilibrium is reached at higher temperature.

3.3 Stability maps for single-phase natural circulation loops

The stability maps represent a helpful tool for the study of the asymptotic

dynamics of natural circulation loops. In general, an equilibrium state can be

identified by means of two dimensionless numbers, such as 𝑆𝑡𝑚-𝑅𝑒, 𝑃𝑟-𝑅𝑒 and

𝑆𝑡𝑚-𝐺𝑟𝑚, where 𝑆𝑡𝑚 is modified Stanton numbers, respectively. In the present

work, the 𝑆𝑡𝑚 and 𝐺𝑟𝑚 numbers have been adopted. They are defined as:

𝑆𝑡𝑚 = 4𝐿𝑡𝐷

𝑁𝑢

𝑅𝑒𝑃𝑟, 𝐺𝑟𝑚 =

𝜌2𝑔𝐷3

𝜇2𝛽Δ𝑇𝑚 (1)

where Δ𝑇𝑚 is a weighted temperature difference inside the loop (for details,

refer to Ruiz et al., 2015 and Vijayan, 2002).

The rigorous procedure followed to compute the stability map is reported in

Cammi et al. (2016), where the effects of thermal inertia are taken into

account. Here only the main steps are briefly summarized.

In defining the governing equations, several assumptions are considered:

The working fluid is considered incompressible and one-dimensional in the

axis direction of the pipes;


71

The Boussinesq approximation is adopted for taking into account variation

of the density due to temperature gradients;

The external heat flux is localized at the heating section;

The heat sink is modelled as a fixed temperature at the cooling section;

The same flow rate and regime are present in the whole loop;

Axial conduction and thermal dissipations in the solid region are

neglected.

With the above simplifications, the governing equations are:

𝜕𝐺

𝜕𝑠= 0 where 𝐺 = 𝜌𝑓

∗𝑈, (2)

𝜕𝐺

𝜕𝑡+𝜕

𝜕𝑠

𝐺2

𝜌𝑓∗ = −

𝜕𝑝

𝜕𝑠−1

2𝑓𝐺2

𝜌𝑓∗

1

𝐷𝑓− 𝑔𝜌𝑓��𝑧 ∙ ��𝑠(𝑠)

with 𝜌𝑓 = 𝜌𝑓∗[1 − 𝛽(𝑇𝑓 − 𝑇𝑓

∗)],

(3)

𝜌𝑓∗𝑐𝑓

𝜕𝑇𝑓

𝜕𝑡+ 𝐺𝑐𝑓

𝜕𝑇𝑓

𝜕𝑠= −ℎ(𝑇𝑓 − 𝑇𝑤,𝑖)

��𝑓

��𝑓, (4)

𝜌𝑤𝑐𝑤𝜕𝑇𝑤,𝑖𝜕𝑡

= ℎ(𝑇𝑓 − 𝑇𝑤,𝑖)��𝑓

��𝑤,𝑖−𝑇𝑤,𝑖 − 𝑇𝑤,𝑜

��𝑤,𝑖��𝑤, (5)

{

𝑇𝑤,𝑜 = 𝑇𝑐 𝑐𝑜𝑜𝑙𝑒𝑟

𝜌𝑤𝑐𝑤𝜕𝑇𝑤,𝑜𝜕𝑡

=𝑇𝑤,𝑖 − 𝑇𝑤,𝑜

��𝑤,𝑜��𝑤+��𝑤,𝑜

��𝑤,𝑜𝑞′′ ℎ𝑒𝑎𝑡𝑒𝑟



��𝑤,𝑜��𝑤 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(6)

The superscript ∗ specifies the reference thermo-physical quantities for the

fluid taken at the cooler entrance, while ��𝑓 , ��𝑤,𝑖, ��𝑤,𝑜 , ��𝑓 , ��𝑤,𝑖, ��𝑤,𝑜 and ��𝑤 are

defined as:

��𝑓 = 𝜋 (𝐷𝑓

2)2

��, ��𝑓 = 𝜋𝐷𝑓��,

��𝑤,𝑖 = 𝜋 [(𝐷𝑤,𝑖2)2

− (𝐷𝑓

2)2

] ��, ��𝑤,𝑖 = 𝜋𝐷𝑤,𝑖𝑠,

��𝑤,𝑜 = 𝜋 [(𝐷𝑤,𝑜2)2

− (𝐷𝑤,𝑖2)2

] ��, ��𝑤,𝑜 = 𝜋𝐷𝑤,𝑜𝑠,

��𝑤 =

ln (𝐷𝑤,𝑜𝐷𝑓

)

2𝜋𝑘𝑤��.

(7)


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72

(a) (b)

Figure 3.4: Discretization of the pipe walls (a) and electrical equivalent model (b).

In this procedure, the pipe wall is discretized along the radial coordinate into

two coaxial shells (Figure 3.4a) by adopting a lumped parameter approach

(Figure 3.4b). Each shell is characterized by its own thermal capacitance (𝑐𝑤)

and a conductive thermal resistance (𝑅𝑤) is placed between them.

In the second step, governing equations are linearized around a steady-state

solution, and the Fourier transform is applied. For the perturbation, the

following form is considered:

𝜙(𝑠, 𝑡) = 𝜙0(𝑠) + 𝛿𝜙(𝑠, 𝑡) = 𝜙0(𝑠) + ��(𝑠)𝑒𝓅𝑡, 𝜔 ∈ ℂ, 𝛿𝜙(𝑠, 𝑡) ≪ 𝜙0(𝑠) (8)

where 𝜙 represents the generic flow variable and 𝓅 the perturbation

pulsation. This strategy not only allows eliding the time derivative, but also

provides a simple linear stability condition for the system, namely the real

part of 𝜔 must be negative. Given a fixed value of 𝐺𝑟𝑚 and defining the loop

geometry and heating distribution, the governing equations can be solved

with the constraint ℛ(𝓅) = 0 and the limit value of 𝑆𝑡𝑚 for which the

equilibrium of the system is stable is found. The collection of the 𝑆𝑡𝑚-𝐺𝑟𝑚

points obtained with this procedure defines, on the stability map, the

transition curve between asymptotically stable and unstable equilibria.

Figure 3.5 shows the stability maps obtained for the L2 loop in the 𝑆𝑡𝑚-𝐺𝑟𝑚

plane both neglecting and considering the wall thermal inertia. In Figure 3.6,

the system equilibria of the different experiments are compared with the

stability maps. The red squares are referred to the experimental unstable

conditions, the green dots to the stable ones. It can be noticed from Figure 3.6

that if the influence of the wall is neglected, the linear analysis is not able to

reproduce the experimental dynamic behaviour while most of the asymptotic

equilibria of the experiments are caught if the effect of thermal inertia are

taken into account. Only three cases are not correctly predicted, namely 7 ℃,

8 ℃ and 9 ℃, which are quite close to the transition curve. In addition, all the

system equilibria are characterised by a 𝐺𝑟𝑚 number in the range of 1011 ÷

1.5 × 1011, corresponding to a 𝑅𝑒 number between 5000 and 6000.


73

Figure 3.5: Stability map for the L2 facility with and without the effect of the piping materials thermal properties (TI

effects).

Figure 3.6: Comparison between the

system equilibria referred to the different experimental cases and the

stability maps.

In this range, the uncertainties on the heat transfer and pressure drop

correlations in the laminar-to-turbulent transition flow regime can affect the

prediction of the stability maps.


The linearization process is based on the assumption that the perturbations

applied to the system are small compared to the steady-state values.

However, the effects of non-linearity can be essential in the more realistic

case of larger perturbations. Furthermore, even though the stability maps

give information on the asymptotic stability, they do not allow predicting the

oscillations modes in case of unstable equilibria. For these reasons, numeric

codes must be adopted.

In the following, the governing equations are solved adopting a CFD

approach. In this way, it is possible to take into account 3D effects and to

avoid the use of empirical correlations for the heat transfer coefficient and the

friction factor, with the drawback of requiring higher computational time

than simpler approaches. The numerical simulations are run by means of the

OpenFOAM® code.

Three different models are adopted to study the dynamic behaviour of the L2

facility. Firstly, the governing equations are solved considering only the

working fluid. Successively, the effect of the pipe wall thermal inertia is

introduced. At last, a complete model for the secondary side of the loop is

considered. Details on these models are reported in sections 3.4.1, 3.4.2 and

3.4.3, respectively, while information about the computational grid is

summarized in section 3.4.4.


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74

3.4.1 Case 1: Water model in OpenFOAM®

Running CFD simulations considering only the working fluid allows studying

simple flow dynamics problems with relative low computational cost, as

observed in the case of the differentially heated buoyant-cavity. However, this

approach often represents only a preliminary step, since the effect of thermal

conduction in solid regions can significantly affect the convection heat

transfer of the fluid in the near-wall region (Cammi et al., 2016). For this

reason simulations are initially run considering only the working fluid and

the conjugate heat transfer problem is taken into account successively.

Thermal properties values of the water at the reference temperature of 300 𝐾

are reported in Table 3.3. 𝜌 is the density, 𝜇 the dynamic viscosity, 𝛽 the

thermal expansion coefficient, 𝑐 the fluid specific heat and 𝑃𝑟 the Prandtl

number. These properties are reported in the transportConstant file in the

constant directory.

Both laminar and turbulent regimes are considered since the flow regime

appears to be in the transition zone, as reported in section 3.3. The flow

regime is specified in the turbulenceProperties file: laminar model is set for


regime. The turbulence model adopted is specified in the RASProperties file.

For this model only the 𝑆𝑆𝑇 𝑘 − 𝜔 model with scalable wall functions is used.

IC and BC for all the flow variables are set in the several files present in the

0-directory. The water is initially at rest at a uniform temperature equals to

room temperature. The no-slip condition is considered as BC for the velocity

while for the pressure the fixedFluxPressure BC is set. The heater is

modelled as a fixed uniform heat flux 𝑞′′ by means of the

turbulentHeatFluxTemperature BC, the cooler as a fixed cold temperature 𝑇𝑐.

The adiabatic condition is imposed at all the other boundaries by applying the

zeroGradient BC for the temperature.

Values for 𝑘 and 𝜔 are chosen using the definition of turbulent kinetic energy

in function of the turbulence intensity and auxiliary expressions adopted for

the specific dissipation:

𝑘 =3

2(𝑈𝑟𝑒𝑓ℐ)

2, 𝜔 = 10

6𝜇

𝒷𝜌𝑦2 (9)

where 𝑈𝑟𝑒𝑓 is a reference value for the velocity, ℐ is the turbulence intensity,

𝒷 is a constant equal to 0.075 and 𝑦 represents the distance of the first node

from the wall. A summary of the IC and BC used is reported in Table 3.4.


75

Table 3.3: Thermal properties of the water at 300 𝐾 (Bergman et al., 2011)

𝝆 (𝒌𝒈 𝒎−𝟑) 𝝁 (𝑷𝒂 𝒔) 𝜷 (𝑲−𝟏) 𝒄 (𝑱 𝒌𝒈−𝟏𝑲−𝟏) 𝑷𝒓

997.01 8.55 × 10−4 2.76 × 10−4 4179 5.83

Table 3.4: Initial and boundary condition adopted for the Water model

IC BC type Velocity (𝒎 𝒔−𝟏) (0,0,0) fixedValue Pressure (𝑷𝒂) 105 fixedFluxPressure

Cooler (𝑻𝒄) - fixedValue Heater (𝒒′′) - turbulentHeatFluxTemperature

Temperature (℃) 20 zeroGradient 𝒌 (𝒎𝟐 𝒔−𝟐) 3.75 × 10−3 kLowReWallFunction 𝝎 (𝒔−𝟏) 146.207 omegaWallFunction

As far as the discretization schemes adopted and the linear equation solvers,

tolerances and algorithm controls setting are concerned, information are

reported in the fvSchemes and the fvSolution files, respectively. As for the

first, upwind scheme is chosen as discretization scheme for divergence terms,

while default schemes are used for the other terms. The tolerance is set to

10−7 for all the physical variables, and the following relaxation factors are

imposed: 0.3 for pressure, 0.7 for velocity and 1 for temperature. The

simulations are run for 10000 𝑠𝑒𝑐 and data outputs are saved every 10 𝑠𝑒𝑐,

instead of 1 𝑠𝑒𝑐 as for the data collection time in the experimental campaign,

because of limit in CPU memory. The buoyantBoussinesqPimpleFoam is

chosen as solver for this case. At last, the maximum Courant number is set at

0.8, in order to obtain more accurate solutions.

3.4.2 Case 2: Pipe Inertia model in OpenFOAM®

In Cammi et al. (2016) and in section 3.3 it has been pointed out the

importance of the wall TI in determining the dynamics of a natural

circulation loop (since for the walls of the system the temperature is not

imposed). Solving the conjugate heat transfer between solid and fluid regions

allows a better prediction of the thermo-physical properties of the working

fluid and consequently of its time-dependent behaviour. For this purpose, the

chtMultiRegionFoam solver is adopted.

Pre-processing setting involves some difference with respect to the previous

model. First of all, solid and fluid regions are defined in the regionProperties

file in the constant directory. In the thermophysicalProperties files it is

necessary to define the additional equation of state used to close the

mathematical system, transport properties and thermal properties values for

the fluid and the solid regions.


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76

Table 3.5: Thermo-Physical Properties (the temperature 𝑇 is expressed in Kelvin).

Water AISI-304 Copper Eq. of state IcoPolynomials rhoConst rhoConst

𝒄 (𝑱 𝒌𝒈−𝟏𝑲−𝟏) 4213 477 385

𝝆 (𝒌𝒈 𝒎−𝟑) −0.4159 𝑇 + 1122 7900 8933

𝝀 (𝑾 𝒎−𝟏𝑲−𝟏) 1.192 × 10−3 𝑇 + 0.2549 14.9 401 𝝁 (𝑷𝒂 𝒔) 1.086 × 10−5 𝑇 + 4.064 × 10−3 - -

As for the working fluid, the icoPolynomials equation of state is adopted. It

allows providing a polynomial dependence on the temperature for the thermal

properties:

𝜙(𝑇)~∑ 𝑎𝑗𝑇𝑗

𝑛

𝑗=0 (10)

where 𝜙 represents the generic thermo-physical property, 𝑛 is the grade of

the polynomial and 𝑎𝑗 are the coefficients. In this way is possible to neglect

the influence of the pressure on the thermal properties. A first order

polynomial is considered for density, thermal conductivity and dynamic

viscosity, while the fluid specific heat is constant in the range of temperature

considered. As for the solid region, both the temperature and the pressure

dependence of the thermo-physical properties are neglected. In Table 3.5, the

interpolating polynomials are reported for the water and the relevant

information on the solid regions is summarized.

As far as the flow regime is concerned, also in this case both laminar and

turbulent regimes are considered and only 𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model is

used.

As for the BC, the turbulentTemperatureCoupledBaffleMixed is adopted for

the solid-fluid interface while the fixed heat flux, the fixed cold temperature

and the adiabatic condition are imposed to the corresponding outer surface of

the pipe wall. A T-junction with inlet/outlet BC for the velocity and a fixed

value for the pressure is present in the left-upper corner of the loop, in order

to model the expansion tank. A detail of the T-junction is shown in Figure 3.7.

The numerical setting for the fluid region is almost the same as the previous

model except for the tolerance in the residual control of the pressure, now set

to 10−5. For the solid regions, Gauss linear discretization scheme is adopted

and a tolerance of 10−6 is considered.


77

3.4.3 Case 3: Heat Exchanger model in OpenFOAM®

In the last model, the secondary side of the loop is modelled taking into

account the complete coaxial cylindrical Heat Exchanger (HE), represented by

an imposed mass flow rate at the external side of the cooling section. In this

way the temperature field is simulated in a more realistic way. Turbulent

flow regime is considered also for the secondary fluid and the 𝑆𝑆𝑇 𝑘 − 𝜔 model

is adopted.

Most of the settings reported for the previous model are still used. The

turbulentTemperatureCoupledBaffleMixed BC is extended also to the

interface secondary fluid-outer pipe wall of the cooler while the fixed

temperature of the heat sink is imposed at the inlet surface of the HE.

zeroGradient BC is applied also at the external surface for the temperature

and at the outlet section of the HE for all the flow quantities except for the

pressure, for which a fixed value equal to the atmospheric pressure is set.

turbulent-Inlet appropriate BC are set for turbulent quantities, fixedValue is

imposed for the velocity and zeroGradient is applied for the pressure at the

inlet section. In Table 3.6 inlet/outlet BC for the HE are reported.

Figure 3.7: T-junction for the expansion tank.

Table 3.6: Inlet/outlet boundary conditions adopted for the HE

In. BC type Out. BC type Velocity fixedValue zeroGradient

Pressure zeroGradient fixedValue Temperature fixedValue zeroGradient

𝒌 turbulentIntensityKineticEnergyInlet zeroGradient 𝝎 turbulentMixingLengthFrequencyInlet zeroGradient


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3.4.4 Mesh generation and independence of the solution

Meshes are generated by means of the ANSYS Workbench® (ANSYS®, 2016)

software and successively converted in the format used in OpenFOAM®. A

sensitivity study is carried out so as to reach the independence of the solution

with respect to the discretization. In this regard, three different meshes are

generated, namely Coarse, Medium and Fine. In Table 3.7, the mesh

sensitivity is reported, including the RE computed taking into account the

mass flow rate at the cooler inlet. Medium grid is chosen to carry out the

simulations for all the cases. Details of the computational grid adopted are

shown in Figures 3.8 and 3.9.

Figure 3.8: Details of the adopted computational grid.

Figure 3.9: Computational grid in the heat-exchanger zone.

Table 3.7: Mesh Sensitivity

Coarse Medium Fine Element Size (𝒎𝒎) 5 4 3.5

Number of elements 156060 259530 361796 RE - 0.55 % 0.01 %

3.5 Numerical simulation results vs. experimental data

79


In this section, the results obtained with the OpenFOAM® code are compared

with the experimental data of the L2 facility provided by IAEA (2014). Only

4 ℃ and 18 ℃ are considered for the heat sink temperature, corresponding to

experimental index 1 and 9 (see Table 3.2), because of the high computational

burden required by CFD simulations. The temperature signals of

thermocouple TC-5 and TC-20 for these two cases are shown in Figures 3.10

and 3.11.

The data reported for the numerical simulations are evaluated by computing

the mean value of the time-dependent radial distribution of the temperature

as follows:

��(𝑡) =1

��(𝑡)𝑆∫ 𝑇(𝑟, 𝑡)𝑢(𝑟, 𝑡)𝑑𝐴𝑆

, ��(𝑡) =1

𝑆∫ 𝑢(𝑟, 𝑡)𝑑𝐴𝑆

(11)

where 𝑆 = 𝜋𝐷2 4⁄ is the cross section of the pipe.

At the end of this section, the best results obtained with the CFD approach

are compared with those obtained with a simpler 1D Object-Oriented (O-O)

model. Details on this model can be found in Cammi et al. (2016).

Figure 3.10: Temperature signals from thermocouple TC-5 and TC-20 for the experiments 1a, 1b, and 1c.


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80

Figure 3.11: Temperature signals from thermocouple TC-5 and TC-20 for the experiment 9.

3.5.1 Case 1: Water model

Figure 3.12 shows the temperature difference across the cooler (Δ𝑇 = 𝑇𝐶-

5 − 𝑇𝐶-20) obtained in case of a heat sink temperature equal to 4 ℃ and the

laminar model. For brevity, only the 1a case is considered. 1b and 1c

experiments show the same dynamics of 1a (see Figure 3.10). Bi-directional

oscillating behaviour is predicted by the model, coherently with the

experimental data. The amplitude of the oscillations is comparable with the

experimental data range but the dynamics is not well caught.

The same result obtained with the turbulence model is reported in Figure

3.13. A slight enhancement can be observed. The amplitude of the oscillations

is smaller than that predicted with the laminar model. Better results are

obtained also for the absolute temperature values, as shown in Figure 3.14.

Figure 3.12: Experimental vs. simulated temperature difference across the cooling section for the experiment 1a obtained neglecting the pipe wall inertia and using the

laminar model.


81

Figure 3.13: Experimental vs. simulated temperature difference across the cooling section for the experiment 1a obtained neglecting the pipe wall inertia and using the

𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.

Figure 3.14: Experimental vs. simulated temperature taken at probe position TC-5 (top) and TC-20 (bottom) for the experiment 1a obtained neglecting the pipe wall

inertia and using the 𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.

Temperature profiles qualitatively follow the same pattern of the

experimental data but the absolute value is slightly underestimated. This is

mainly due to the simplify assumptions adopted in this model.

Figures 3.15 and 3.16 show the temperature difference obtained for the

transient at 18 ℃ with the laminar and turbulent flow regimes, respectively.

In this case, the stable behaviour of the fluid flow is not predicted by both the

models.


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82

Figure 3.15: Experimental vs. simulated temperature difference across the cooling section for the experiment 9 obtained neglecting the pipe wall inertia and using the

laminar model.

Figure 3.16: Experimental vs. simulated temperature difference across the cooling section for the experiment 9 obtained neglecting the pipe wall inertia and using the


3.5.2 Case 2: Pipe Inertia model

In this section, the results obtained considering the effect of thermal inertia

and applying an imposed wall temperature BC at the cooling section are

reported. The results of the laminar simulations are not presented since

they do not reproduce the experimental behaviour, as observed also in

section 3.5.1. Due to the higher computational cost required by the

chtMultiRegionFoam solver, the simulations are run for 5000 𝑠.

Figures 3.17 and 3.18 illustrate the temperature difference of the working

fluid across the cooling section and the absolute temperature values for the 1a

case, respectively.


83

Figure 3.17: Experimental vs. simulated temperature difference across the cooling section for the experiment 1a obtained considering the pipe wall inertia and using

the 𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.

Figure 3.18: Experimental vs. simulated temperature taken at probe position TC-5 (top) and TC-20 (bottom) for the experiment 1a obtained considering the pipe wall


The temperature difference shows a better agreement, with oscillations

characterized by smaller amplitude than the previous model, while the

absolute temperature values are underestimated. It can be explained by the

choice of imposing a fixed temperature at the outer surface of the cooler for

the heat sink rather than considering the complete model of the HE. This

leads to a lower temperature field at the inner surface of the cooler pipe wall

and, consequently, to a lower temperature field of the working fluid.

The results obtained for the transient at 18°C are reported in Figures 3.19

and 3.20. This time, the dynamic behaviour is caught by the model, as well as

the temperature difference at the cooling section. On the contrary, the

absolute value of the temperature is again underestimated.


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84

Figure 3.19: Experimental vs. simulated temperature difference across the cooling section for the experiment 9 obtained considering the pipe wall inertia and using the


Figure 3.20: Experimental vs. simulated temperature taken at probe position TC-5 (top) and TC-20 (bottom) for the experiment 9 obtained considering the pipe wall


Table 3.8: Steady-state values of temperature

𝚫𝑻 𝑻𝒊𝒏 𝑻𝒐𝒖𝒕 Experiment 9 −6.97 ℃ TC-20= 67.61 ℃ TC-5= 60.64 ℃

Pipe Inertia model −7.80 ℃ TC-20= 47.33 ℃ TC-5= 39.53 ℃ RE 11.908 % 34.82 % 29.99 %

Steady-state values of temperature difference and absolute temperatures are

reported in Table 3.8.

At last, a comparison between the results of the water model and the pipe

inertia model is reported in Figures 3.21 and 3.22 for the case at 18 ℃, in

order to show the improvement obtained in predicting the dynamic behaviour

of the system.


85

Figure 3.21: Simulated temperature difference across the cooling section for the experiment 9 with and without the TI effect obtained using the 𝑆𝑆𝑇 𝑘 − 𝜔 turbulence

model.

Figure 3.22: Simulated temperature taken at probe position TC-5 (top) and TC-20 (bottom) for the experiment 9 with and without TI effect obtained using the


3.5.3 Case 3: Heat Exchanger model

The results obtained with the previous model are in agreement with the

stability map but the temperature field predicted are lower than the

corresponding experimental data (index 1a and 9). To overcome this problem,

simulations are run modelling the heat sink with a coaxial heat exchanger as

in the real experiments.

Figures 3.23 and 3.24 show the comparison between the numerical

simulations and the experimental data with a heat sink temperature of 4 ℃,

Figures 3.25 and 3.26 for the case at 18 ℃.


loops

86

Figure 3.23: Experimental vs. simulated temperature difference across the cooling section for the experiment 1a obtained with HE model and using the 𝑆𝑆𝑇 𝑘 − 𝜔

turbulence model.

Figure 3.24: Experimental vs. simulated temperature taken at probe position TC-5 (top) and TC-20 (bottom) for the experiment 1a obtained with HE model and using

the 𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.

Table 3.9: Mean values of temperature in the time interval 2030 𝑠-2400 𝑠

𝚫𝑻 𝑻𝒊𝒏 𝑻𝒐𝒖𝒕 Experiment 1a 8.65 ℃ TC-5= 59.46 ℃ TC-20= 50.81 ℃

HE model −7.93 ℃ TC-20= 50.23 ℃ TC-5= 42.30 ℃ RE 8.324 % 15.527 % 16.759 %

The simulations correctly predict the experimental dynamic behaviour for all

the transients. Both the temperature difference across the cooler section and

the absolute values of the temperature are in good agreement with the

experiments, but a slight underestimation persists.

In Tables 3.9 and 3.10, the comparison between the mean values of

experimental and computed temperature are reported for the two cases.


87

Figure 3.25: Experimental vs. simulated temperature difference across the cooling section for the experiment 9 obtained with HE model and using the 𝑆𝑆𝑇 𝑘 − 𝜔

turbulence model.

Figure 3.26: Experimental vs. simulated temperature taken at probe position TC-5 (top) and TC-20 (bottom) for the experiment 9 obtained with HE model and using the


Table 3.10: Steady-state values of temperature

𝚫𝑻 𝑻𝒊𝒏 𝑻𝒐𝒖𝒕 Experiment 9 −6.97 ℃ TC-20= 67.61 ℃ TC-5= 60.64 ℃ HE model 7.53 ℃ TC-5= 61.17 ℃ TC-20= 53.64 ℃

RE 8.034 % 9.525 % 11.563 %

In case of unstable behaviour, it is important to underline that the

experimental oscillation modes cannot be numerically reproduced. This is

mainly due to the chaotic nature of natural circulation instabilities. Furthermore, it

is well known that, from an experimental point of view, when the same

experiment is repeated several times keeping fixed all the parameters, the

shape and the amplitude of the oscillations are quite the same while the

phase is usually different between the different acquisitions. This fact is


loops

88

clearly shown in Figure 3.10. From Tables 3.9 and 3.910, it can be noticed an

opposite signs in the temperature difference. This is due to the different

direction in which the experimental and simulated natural circulation flows

have been established. When the locations of heater and cooler are

symmetric, as the case of the L2 facility, there is not any preferred direction

of the fluid flow and the inception of either clockwise or anticlockwise motion

depends on small random external disturbances. This fact is shown not only

by the numerical simulations but also by the several experimental runs, as

illustrated in Figures 3.10.

From the achieved results, the modelling of the coaxial heat exchanger is

necessary to obtain the correct temperature distribution inside the system

and to reproduce the experimental data.

As mentioned in Section 3.2, the unstable behaviour inside the L2 facility can

be ascribed to the formation of hot and cold plugs inside the system, according

to Vijayan (2002). When the system presents an oscillating behaviour,

acceleration and stagnation of the fluid flow occur. During the rest condition,

the temperature of the working fluid in the heater increases leading to the

formation of a hot plug (similarly, in the cooler a cold plug is formed).

Gradually, the hot plug becomes larger, reaches the vertical sections of the

loop and when its rise into one of the vertical legs is more than the other (e.g.,

due to some small external perturbations), the circulation starts. As the hot

plug goes up, the flow accelerates thanks to the increment of the buoyancy

force and has a peak when the entire hot plug is in the vertical rising leg.

Then, the hot plug passes in the top horizontal leg and enters in the vertical

(descending) opposite leg with a drop of the buoyancy acceleration and hence

of the flow. The reduction of the circulation continues until the buoyancy force

contribution of the rising vertical leg becomes smaller than that of the

descending one. At this point, part of the hot plug returns in the upper

horizontal leg leading to flow stagnation, while the coldest fluid in the loop

occupies a portion of the lower horizontal leg, which acts like a check valve

that only allows forward circulation. After the stop of the circulation, other

hot and cold plugs are formed in the heater and cooler, respectively. Minor

forward and backward oscillations take place as the hot plug in the heater

attempts to reach one of the vertical legs. If the main flow during these minor

peaks is sufficient to move the hot plug closer to one of vertical legs

alternatively, bi-directional oscillating behaviour is observed. Otherwise, the

process repeats itself giving unidirectional pulsing flow. In this regard, the 3D

CFD approach is useful to highlight the hot/cold plug formation process.

In Figures 3.27÷3.32, the instantaneous temperature field taken at different

time intervals during a mass flow rate inversion is shown. In Figure 3.27,

during the rest flow condition, the formation of the hot and cold plugs is


89

occurring at the heater and the cooler, respectively. Then, the plugs start

sliding towards the vertical legs (Figure 3.28) until they occupy the entire

section (Figure 3.29). At this point, the plugs enter in the horizontal sections

leading to a reduction of the buoyancy force (Figure 3.30) and unidirectional

oscillations characterise the fluid flow. Progressively, the amplitude of such

oscillations increases until a mass flow reversal occurs (Figure 3.31). The

described process shows a recurring pattern triggered by the inversion of the

mass flow rate (Figures 3.31 and 3.32). In this regard, Figures 3.33 and 3.34

show the inversion of direction of the circulation in terms of the velocity field.

Figure 3.27: CFD temperature distribution at 1970 𝑠, hot (cold) plug is developing.

Figure 3.28: CFD temperature distribution at 1990 𝑠, sliding of the hot (cold) plug in

the right (left) vertical leg.

Figure 3.29: CFD temperature distribution at 2000 𝑠, the hot (cold) plug occupies the

entire right (left) vertical leg.

Figure 3.30: CFD temperature distribution

at 2010 𝑠, the hot (cold) plug starts to enter in the left (right) vertical leg.


loops

90

Figure 3.31: CFD temperature distribution at 2330 𝑠, mass flow rate inversion from

the anticlockwise direction to the clockwise one.

Figure 3.32: CFD temperature distribution

at 2430 𝑠, mass flow rate inversion from the clockwise direction to the

anticlockwise one.

Figure 3.33: Distribution of the x component of the velocity (𝑈𝑥) at 2330 𝑠

(clockwise circulation).

Figure 3.34: Distribution of the x

component of the velocity (𝑈𝑥) at 2430 𝑠 (anticlockwise circulation).

In addition to the hot/cold plugs formation, the CFD approach allows catching

also the radial distribution of the flow fields. An example of the temperature

and the velocity distributions inside the heater and the cooler sections is

reported in Figures 3.35÷3.38.


91

Figure 3.35: CFD radial temperature distribution taken at TC-20 position

(1990 𝑠).

Figure 3.36: CFD radial temperature distribution taken at TC-10 position

(1990 𝑠).

Figure 3.37: CFD radial velocity

magnitude distribution taken at TC-20 position (1990 𝑠).

Figure 3.38: CFD radial velocity

magnitude distribution taken at TC-10 position (1990 𝑠).

In order to understand the influence of the turbulence model on the results,

the same simulations have been repeated with the realizable 𝑘 − 휀 model.

From Figures 3.39 and 3.40, it is visible that an unstable dynamic behaviour

is predicted with a heat sink temperature of 18 ℃, while the amplitude of the

oscillations is acceptable with respect to the experimental data range. Hence,

the sensitivity of the results with respect to the turbulence model, that is a

key issue of any CFD simulation, is even more crucial for natural circulation.

In this regard, the 𝑆𝑆𝑇 𝑘 − 𝜔 with scalable wall functions seems to be a good

choice. This result does not surprise if we consider the strengths and

weaknesses of the two turbulence models adopted (Wilcox, 1993b; Versteeg &

Malalasekera, 2007), but it is fundamental to extend the investigation

adopting advanced turbulence models, such as non-linear two equations

models or the Reynolds Stress Model (seven equations model).


loops

92

Figure 3.39: Experimental vs. simulated temperature difference across the cooling section for the experiment 1a obtained with HE model and using the realizable 𝑘 − 휀

turbulence model.

Figure 3.40: Experimental vs. simulated temperature difference across the cooling section for the experiment 9 obtained with HE model and using the realizable 𝑘 − 휀

turbulence model.

3.5.4 CDF model vs. O-O model

In this section, the results obtained with the HE model are compared with

those of the 1D O-O model developed in Cammi et al. (2016) for the two cases

considered in this work. The O-O approach is based on the Modelica language

and it is implemented in the Dymola simulation environment. In this model,

the governing equations are discretized by means of the finite element

method and the correlations presented in Ruiz et al. (2015) and Pini et al.

(2016) are used for the modelling of the heat exchange and the friction losses.

The results are summarized in Figures 3.41÷3.46.


93

Figure 3.41: Experimental vs. simulated temperature (O-O and CFD with 𝑆𝑆𝑇 𝑘 −𝜔) taken at probe position TC-20 for

experiment 1a.

Figure 3.42: Experimental vs. simulated

temperature (O-O and CFD with 𝑆𝑆𝑇 𝑘 − 𝜔) taken at probe position TC-5

for experiment 1a.

Figure 3.43: Experimental vs. simulated temperature (O-O and CFD with 𝑆𝑆𝑇 𝑘 −𝜔) taken at probe position TC-20 for

experiment 9.


temperature (O-O and CFD with 𝑆𝑆𝑇 𝑘 − 𝜔) taken at probe position TC-5

for experiment 9.

Figure 3.45: Experimental vs. simulated (O-O and CFD with 𝑘 − 𝜔 ) temperature difference across the cooling section for

the experiment 1a.


(O-O and CFD with 𝑆𝑆𝑇 𝑘 − 𝜔) temperature difference across the cooling

section for the experiment 9.

The approaches give very similar results that are in agreement with the

experimental data. For stable transients both the CFD and O-O predictions

converge to quite similar values of the absolute temperature difference

(7.53 ℃ and 6.66 ℃, respectively), while for the unstable transients the

oscillation modes are comparable. On one hand, the O-O analysis is able to

reproduce the dynamic behaviour of natural circulation although it relies on


loops

94

simplified governing equations (with respect to CFD) and empirical

correlations for the pressure drops (distributed and localised) and the heat

exchange are adopted. On the other hand, the CFD approach is necessary to

highlight the 3D effects that can occur in this kind of systems (e.g., the

hot/cold plug formation and radial distributions of the flow quantities) and it

can correctly simulate the trigger mechanism of natural circulation, while the

O-O model cannot start from the stagnant condition of the flow (Pilkhwal et

al., 2007). Moreover, while the further improvement of the O-O model is quite

limited, for the CFD approach room of improvement would be possible.

3.6 Final remarks

In this chapter, the results obtained with the OpenFOAM® code are compared

with the experimental data provided by IAEA (2014), in order to validate the

models developed for the study of natural circulation dynamic behaviour in

rectangular loops. A satisfactory agreement is achieved.

From the obtained results, it can be underlined that for the prediction of the

dynamics of natural circulation loops the effect of the pipe wall thermal

inertia is fundamental (the temperature of the wall is not imposed, differently

from the cavity case), as well as the correct evaluation of the physical

properties of the working fluid. In this regard, it is necessary to adopt a

complete model for the cooling section. If a fixed wall-temperature is imposed

at the external surface of the cooler, the computed temperature field is lower

than the experimental one, while the adoption of a complete model for the

heat exchanger allows a better prediction of the temperature. Anyway, a

slightly underestimation persists also when the coaxial heat exchanger is

taken into account.

The CFD is able to correctly simulate the triggering of natural circulation,

starting from the rest condition of the flow, and is useful to highlight the 3D

effects that can occur in this kind of systems (e.g., the radial temperature and

velocity distributions). However, the quality of the CFD simulations strongly

depends on the choice of the turbulence models. Improvements in the results

can be achieved by adopting advanced turbulence models, such as non-linear

two equations models or the Reynolds Stress Model (seven equations model),

but this strategy will substantially increase the required computational

resource.

References

95

References

ANSYS, 2016, http://www.ansys.com/it.



NJ, United States.




Chen, K., 1985. On the instability of closed-loop thermosiphons. Journal of

Heat Transfer, Issue 107, pp. 826-832.

IAEA, 2014. Progress in Methodologies for the Assessment of Passive System

Reliability in Advanced Reactors. Vienna, IAEA-TECDOC-1752.

Keller, J. B., 1966. Periodic oscillations in a model of thermal convection.

Journal of Fluid Mechanics, 1(26), pp. 599-606.

Misale, M., 2014. Overview on single-phase natural circulation loops. Phuket,

Thailand, s.n.

Misale, M. & Garibaldi, P., 2010. Dynamic behaviour of a rectangular singl-

phase natural circulation loop: influence of inclination.. Mumbai, India, s.n.

Misale, M., Ruffino, P. & Frogheri, M., 2000. The influence of the wall

thermal capacity and axial conduction over a single-phase natural circulation

loop: 2-D numerical study. Heat and Mass Transfer, Issue 36, pp. 533-539.


Pilkhwal, D. et al., 2007. Analysis of the unstable behaviour of a single-phase

natural circulation loop with one-dimensional and computational fluid-

dynamic models. Annals of Nuclear Energy, Issue 34, pp. 339-355.



with internally heated fluids.. Chemical Engineering Science, Issue 145, pp.

108-125.



Chemical Engineering Science, Issue 126, pp. 573-583.

http://www.ansys.com/it




loops

96

Swapnalee, B. P. & Vijayan, P. K., 2011. A generalized flow equation for

single phase natural circulation loops obeying multiple friction laws.

International Journal of Heat and Mass Transfer, Issue 54, pp. 2618-2629.



Vijayan, P., 2002. Experimental observations on the general trends of the

steady state and stability behaviour of single-phase natural circulation loops.

Nuclear Engineering and Design, Issue 215, pp. 139-152.

Vijayan, P. K., Austregesilo, H. & Teschendorff, V., 1995. Simulation of the

unstable oscillatory behaviour of single phase natural circulation with

repetitive flow reversals in a rectangular loop using the computer code

ATHLET. Nuclear Engineering and Design, Issue 155, pp. 614-623.

Vijayan, P. K., Sharma, A. K. & Saha, D., 2007. Steady State and stability

characteristics of single-phase natural circulation in a rectangular loop with

different heater and cooler orientations. Experimental Thermal and Fluid

Science, Issue 31, pp. 925-945.

Welander, P., 1967. On the oscillatory instability of a differentially heated

fluid loop. Journal of Fluid Mechanics, Issue 29, pp. 17-30.


Industries Inc.

99

CHAPTER 4: Dynamic stability for natural

circulation with internally heated fluid: the

DYNASTY facility

4.1 Introduction

One of the main peculiarities of natural circulation systems is the possibility

to avoid the use of active components for emergency systems, ensuring a

higher degree of reliability. In the last years, more and more attention has

been paid to natural convection with fluids characterized by a volumetric heat

source, especially for innovative applications in nuclear energy field, whose

main example is given by the Generation IV MSR (Luzzi et al., 2010; Fiorina

et al., 2014; GIF, 2014; Srivastava et al., 2015; Serp et al., 2016). In this

system, the working fluid is a molten salt in which the nuclear fuel is directly

dissolved. As a consequence, besides fission reactions in the core region, a

distributed volumetric heat source is also present in the primary circuit, due

to the decay of fission products.

The presence of IHG can affect the dynamics of natural circulation with

respect to the conventional case (i.e., without IHG) and may lead to undesired

behaviours. Since one of the key features in the design of Generation IV

nuclear reactors is to guarantee a high level of intrinsic safety, the

SAMOFAR Project (SAMOFAR, 2015) includes a detailed study of such

problems by means of advanced numerical codes and experimental

investigations.

Even though many studies have been conducted on natural circulation loops,

only few works include the case of the IHG. In Pini et al. (2014) and Ruiz et

al. (2015), a semi-analytical linear stability approach is developed neglecting

the effect of mass flow oscillation on the convective heat transfer coefficient.

Successively, Pini et al. (2016) introduce the heat exchange effect on the

dynamics, taking into account also the effect of non-linearity by adopting a

numerical approach and comparing the results with the information provided

by the stability maps. In Cammi et al. (2016), the studies of the previous

works are extended including also the effect of piping materials. However, the

methods developed in these works are purely methodological. In this regard,

the DYNASTY (DYnamics of NAtural circulation for molten SalT internallY

heated) testing facility is under construction at Energy Labs of Politecnico di

Milano in order to provide the necessary experimental data for the validation

CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:

the DYNASTY facility

100

of these methods. In this contest, it is of interest to underline that the design

procedure of the facility has been based on the analysis tools developed in

these works and validated in case of conventional natural circulation.

In this chapter, some operative transients of the DYNASTY facility are

simulated by means of the OpenFOAM® code, and the results are compared

with those obtained by means of the numerical model developed in Cammi et

al. (2016).

As far as the structure of this chapter is concerned, a general description of

the DYNASTY facility is given in the first section, and some information

about the working fluid is reported in section 4.2. Successively, the semi-

analytical stability analysis is briefly summarized in section 4.3. Section 4.4

deals with the numerical implementation, while the obtained results are

presented in section 4.5. In the last section, the main conclusions are drawn.

4.2 Description of the DYNASTY facility

The scheme of the DYNASTY loop is shown in Figure 4.1 while geometric

data of the loop are reported in Table 4.1. 𝐷, 𝑊 and 𝐻 are the inner diameter,

the width and the height of the rectangular loop, respectively. 𝓉 is the

thickness of the pipe wall, 𝐿𝑡𝑜𝑡 is the total length, and 𝐻 𝑊⁄ is the aspect ratio.

Figure 4.1: DYNASTY scheme (not in scale).

4.2 Description of the DYNASTY facility

101

Table 4.1: DYNASTY dimensions

𝑫 (𝒎𝒎) 𝓽 (𝒎𝒎) 𝑾 (𝒎𝒎) 𝑯 (𝒎𝒎) 𝑳𝒕𝒐𝒕 (𝒎𝒎) 𝑳𝒕𝒐𝒕 𝑫⁄ 𝑯 𝑾⁄

38.2 2 2400 3200 11200 293.19 1.33

The experimental rig consists of a vertical rectangular loop with a circular

cross-section of constant diameter. The structural components and pipes are

made of stainless steel (AISI-304 and AISI-316) while a molten salt is chosen

as working fluid. Details on the working fluid are reported in the next section.

As for the heating mode, in general the distributed volumetric heat source

can be experimentally induced by forcing an electrical current to flow into the

fluid, but this option involves some technical issues, e.g. the necessity to

ensure electrical insulation in order to avoid both electrochemical phenomena

and eddy currents. For this reason, in the DYNASTY facility the induction of

the internal energy source is realised by means of an All-External Heat Flux

(A-EHF) homogeneously distributed along the entire loop, except for the

cooler section. In Cammi et al. (2016), it has been highlighted that for natural

circulation loops characterised by a length-to-diameter ratio (𝐿𝑡𝑜𝑡 𝐷⁄ ) very

high, the A-EHF is a good approximation of the IHG, which is the case of the

DYNASTY facility loop (see Table 4.1). Some differences can be observed in

the high 𝑅𝑒 region and can be due to the fact that in the cooler section the

heat flux is not applied in case of the A-EHF. The heat flux is provided by

means of fiberglass electrical resistances in a range of 0.5 ÷ 10 𝑘𝑊. The power

line is divided into four independent groups in order to allow several heating

set-ups, from localized heating in different power sections to run conventional

natural circulation experiments, to the A-EHF for distributed heating. In

addition, it is also possible to combine localized and distributed heating

modes. The cooling is realised with a finned pipe in the top horizontal leg of

the loop and it is possible to operate in coupled mode with an axial fan able to

provide a volumetric flow rate up to 5000 𝑚3 ℎ−1.

The bottom part of the loop presents two branches in order to run specific

experiment types: the top one includes a centrifugal pump completely made of

AISI-316 whose purpose is to initialize the mass flow at system start-up, and

allows conducting experiments in forced convection; the bottom branch is

equipped with a flow-meter and is used for natural circulation experiments.

The tank in the top part of the loop is used for filling up the loop but it also

serves as expansion tank, in order to allow working fluid to expand,

maintaining the internal pressure equal to the atmospheric one. A second

tank is placed at the bottom of the loop for the salt storage during the

draining procedure. Accurate measures of the temperature of both the solid

and the fluid regions are provided by several thermocouples, which are

insulated with mineral wool material.



102

At last, some information about the DYNASTY operative range is provided.

The maximum mass flow rate achievable in natural circulation mode is

0.35 𝑘𝑔 𝑠−1, corresponding to a 𝑅𝑒 of 4500, while in forced convection is

4 𝑘𝑔 𝑠−1 (𝑅𝑒 = 10000). The minimum and maximum operative temperatures

are 523 𝐾 and 623 𝐾, respectively.

4.2.1 Hitec® molten salt

The study of the natural circulation of molten salts is of interest not only for

innovative nuclear reactor concepts, but also in many other applications

(Srivastava et al., 2015), e.g. for transport and thermal energy storage in

high-efficiency solar power plants. Molten salts are characterized by several

properties which make them an interesting choice as coolant, such as a high

volumetric heat capacity and boiling temperature, which allow for reducing

the volume of salt required and for avoiding overpressure requirements,

respectively, and in nuclear energy applications they can be able to dissolve

actinides and to resist to radiations. Besides these advantages, other

properties must be taken into account in the selection of the salt, namely

thermal stability at high temperature, low melting point, compatibility with

other materials, availability and costs. An overview of the most common

molten salts considered and their properties can be found in Serrano-Lo pez et

al. (2013).

In the DYNASTY facility, the salt chosen as working fluid is a mixture

commercially known as Hitec®, whose main application field is in solar

systems for power generation. It is composed of 𝑁𝑎𝑁𝑂3 (7𝑤𝑡%),

𝑁𝑎𝑁𝑂2 (40𝑤𝑡%) and 𝐾𝑁𝑂3 (53𝑤𝑡%), with a melting point of 415 𝐾 and it can

be considered thermally stable up to ~873 𝐾 in inert atmosphere, but this

limit can be extended between 923 𝐾 and 973 𝐾 in an oxidising atmosphere.

More information about thermal stability of nitrites/nitrates salts are

reported in Olivares (2012).

As far as the thermo-physical properties are concerned, expression reported

in Serrano-Lopez et al. (2013) are considered. Density is the best known

property, showing very small differences among the several correlations

founded in literature, while more problems arise in defining correlations for

dynamic viscosity, thermal conductivity and specific heat. In particular, for

the last two properties a constant representative value is suggested, because

of the disagreement between the data reported in several works. Figure 4.2

shows the values of density and dynamic viscosity of the Hitec® molten salt as

a function of temperature, whose temperature range of validity is (448 ÷

773) 𝐾 and (525 ÷ 773) 𝐾, respectively.

4.3 Stability maps for single-phase natural circulation loops with internally

heated fluid

103

Figure 4.2: Hitec® density (top) and dynamic viscosity (bottom) as a function of

temperature.


with internally heated fluid

In the formulation of stability maps for single-phase natural circulation loops,

the presence of the IHG has been introduced only in recent works (Pini et al.,

2014; Ruiz et al., 2015; Pini et al., 2016; Cammi et al., 2016). In chapter 3, the

procedure followed to compute the stability map is briefly summarized in case

of conventional natural circulation. Here, only the main differences are

highlighted.

The additional assumption made in defining the governing equations is to

consider the IHG (𝑞′′′) as a homogeneously distributed energy source along

the entire loop. From the considerations reported in the previous section,

however, an A-EHF (𝑞#′′) is adopted in order to mimic the effect of IHG. For

this reason, it is necessary to modify the governing equations. In particular,

the 𝑞′′′ term is no longer considered into the energy equation of the fluid (Eq.

(1)), while the 𝑞#′′ term is introduced in the energy equation of the outer shell

of the pipe wall (Eq. (2)):



104

IHG A-EHF

𝜌𝑓∗𝑐𝑓

𝜕𝑇𝑓


𝜕𝑇𝑓

𝜕𝑠=

= −ℎ(𝑇𝑓 − 𝑇𝑤,𝑖)��𝑓

��𝑓+ 𝑞′′′

𝜌𝑓∗𝑐𝑓

𝜕𝑇𝑓


𝜕𝑇𝑓

𝜕𝑠= −ℎ(𝑇𝑓 − 𝑇𝑤,𝑖)

��𝑓

��𝑓 (1)

{

𝑇𝑤,𝑜 = 𝑇𝑐 (𝑎)



��𝑤,𝑜��𝑤+��𝑤,𝑜

��𝑤,𝑜𝑞′′ (𝑏)



��𝑤,𝑜��𝑤 (𝑐)

{

𝑇𝑤,𝑜 = 𝑇𝑐 (𝑎)



��𝑤,𝑜��𝑤+��𝑤,𝑜

��𝑤,𝑜𝑞′′ (𝑏)



��𝑤,𝑜��𝑤+ 𝑞#

′′ (𝑐)

(2)

where 𝑎, 𝑏 and 𝑐 are referred to the cooler, the heater and otherwise region,

respectively.

In Figure 4.3, the stability maps of the DYNASTY facility in the 𝑆𝑡𝑚-𝐺𝑟𝑚

plane are shown for three different heating modes, namely IHG, A-EHF and

Localized Heat Flux (LHF). In particular, for the latter case a localized heater

is considered on the left vertical leg of the loop, in a Vertical-Heater

Horizontal-Cooler (VHHC) configuration. In this way, there is preferential

direction of the flow, due to the asymmetry of the system with respect to the

vertical axis (see Figure 4.1), promoting the stability of the system. When the

IHG (or the A-EHF) case is considered, the system is characterized by a

perfect axial-symmetry and the fluid has not a preferred flowing direction

anymore. This lack leads to an increase of the instability, especially at low 𝑅𝑒.

Figure 4.3: DYNASTY stability map in the 𝑆𝑡𝑚-𝐺𝑟𝑚 plane for the IHG, A-EHF and

LHF cases.


105

The results depicted in Figure 4.3 confirm that the A-EHF and the IHG

induce a comparable effect on the dynamic stability of natural circulation

loops characterised by a very high 𝐿𝑡𝑜𝑡 𝐷⁄ ratio, as shown also in Cammi et al.

(2016).


Based on the information of the asymptotic equilibrium given by the stability

map, the time-dependent behaviour of the system is studied by means of CFD

simulations. From the considerations reported in Cammi et al. (2016) and in

chapter 3, the simulations are run considering the effect of the piping

materials.

Also in this case, meshes are generated by means of the ANSYS Workbench®

(ANSYS®, 2016) software and successively converted in the format used in

OpenFOAM®, and a sensitivity study is performed using three different

meshes, namely: Coarse, Medium and Fine. The simulations are run

considering an imposed LHF on the left vertical leg of the loop. The results

obtained are summarized in Table 4.2, showing the size and number of the

elements, and the RE between two successive values of the mass flow rate

computed at the left section of the cooler.

Due to the high computational cost required by CFD simulations with meshes

characterized by a number of elements greater of 106, Fine grid is chosen to

carry out the simulations for all the cases, showing an acceptable RE, less

than 2 %. A detail of the computational grid adopted is shown in Figure 4.4.

Figure 4.4: Details of the adopted computational grid.

Table 4.2: Mesh Sensitivity

Coarse Medium Fine Element Size (𝒎𝒎) 5.5 5 4.5

Number of elements 588558 655324 885959 RE - 2.804 % 1.724 %



106

A T-junction with inlet/outlet BC for the velocity and a fixed value for the

pressure is present in the left-upper corner of the loop, in order to model the

expansion tank.

Thermo-physical properties of the Hitec® molten salt are defined using the

correlations reported in Serrano-Lopez et al. (2013) and adopting the

icoPolynomials as equation of state. The correlation given for the dynamic

viscosity (see Figure 4.2) is interpolated using a fourth order polynomial. A

first order polynomial is proposed for the density, and constant values are

used for the thermal conductivity and the fluid specific heat. As for the solid

region, both the temperature and the pressure dependence of the thermo-

physical properties are neglected, and AISI-304 is considered as piping

material. In Table 4.3, the interpolating polynomials are reported for the salt

and the relevant information on the solid regions is summarized.

As far as the flow inside the loop is concerned, laminar flow regime is

considered for lower thermal power levels, while higher power levels are in

the transition region, so the 𝑆𝑆𝑇 𝑘 − 𝜔 model with scalable wall functions is

adopted.

As for the IC and BC, the fluid is initially at rest at a uniform temperature

equals to 523 𝐾. The same initial temperature is assumed for the solid region.

At the solid-fluid interface, the no-slip condition is considered as BC for the

velocity and the turbulentTemperatureCoupledBaffleMixed and the

fixedFluxPressure BC are adopted for the temperature and the pressure,

respectively. In the A-EHF case, the fixed uniform heat flux 𝑞#′′ is modelled by

means of the turbulentHeatFluxTemperature BC. In the IHG case, an

internal energy source 𝑞′′′ is considered in the fluid region and the adiabatic

condition is imposed at all the outer surfaces of the pipe wall by applying the

zeroGradient BC for the temperature field. In both cases, the cooler is

modelled as a fixed cold temperature 𝑇𝑐.

The simulations are run for 2500 𝑠𝑒𝑐 and data outputs are saved every 10 𝑠𝑒𝑐,

because of the high computational cost of the simulations and the limit in

CPU memory. All the other numerical settings are the same as those adopted

for the Pipe Wall model in Section 3.4.2.

Table 4.3: Thermo-Physical Properties (the temperature 𝑇 is expressed in Kelvin)

Hitec® AISI-304 Eq. of state icoPolynomials rhoConst

𝒄 (𝑱 𝒌𝒈−𝟏𝑲−𝟏) 1560 477

𝝆 (𝒌𝒈 𝒎−𝟑) −0.7324 𝑇 + 2279.799 7900

𝝀 (𝑾 𝒎−𝟏𝑲−𝟏) 0.48 14.9

𝝁 (𝑷𝒂 𝒔) 1.696577 × 10−12 𝑇4 − 4.482646 × 10−9 𝑇3 +

+4.478099 × 10−6 𝑇2 − 2.014653 × 10−3 𝑇 + 0.3477632 -

4.5 Results

107

4.5 Results

In this section, the results obtained for some transients of the DYNASTY

facility loop are reported. In particular, three different power level are chosen,

namely 10 𝑘𝑊, 2 𝑘𝑊 and 0.5 𝑘𝑊, and both the A-EHF and IHG heating modes

are considered.

These operative transients are previously simulated with the 1D O-O model

developed in Cammi et al. (2016), showing a good agreement with the the

information provided by the stability map, as illustrated in Figure 4.5. The

stable equilibria are represented with a green dot, while blu squares indicate

the unstable ones.

For the CFD simulations, the mass flow rate is computed by evaluating the

mean value of the time-dependent radial distribution of the temperature and

velocity as follows:

��(𝑡) =1

��(𝑡)𝑆∫ 𝑇(𝑟, 𝑡)𝑢(𝑟, 𝑡)𝑑𝐴𝑆

, ��(𝑡) =1

𝑆∫ 𝑢(𝑟, 𝑡)𝑑𝐴𝑆

(3)

and successively applying the definition of mass flow rate:

Γ(𝑡) = 𝜌(��(𝑡))𝑆��(𝑡) (4)

Where Γ is the mass flow rate and 𝑆 = 𝜋𝐷2 4⁄ is the cross section of the pipe.

In Figure 4.6, the results obtained with the OpenFOAM® code for the case at

10 𝑘𝑊 are depicted, while Figure 4.7 shows the comparison with the solutions

of the 1D O-O model.

Figure 4.5: Comparison of the asymptotic equilibria obtained with the 1D O-O model and the stability map, for the operative transients considered.



108

Figure 4.6: Mass flow rate transient behaviour obtained with the OpenFOAM® code for the IHG and A-EHF cases, at 10 𝑘𝑊.

(a)

(b)

Figure 4.7: CFD vs. O-O models for the IHG (a) and A-EHF (b) heating modes, at

10 𝑘𝑊.

A stable behaviour is predicted in both the A-EHF and IHG situations,

coherently with the information provided by the stability map. The only

differences that can be noticed between the two heating modes are a slower

dynamics and a slight overestimation of the mass flow rate for the A-EHF

case. When heat is provided by means of an imposed external heat flux, the

energy is firstly absorbed by the piping materials, as a consequence of the

4.5 Results

109

capacitive effect of the thermal inertia. This energy is successively yielded to

the fluid, starting to rise the temperature of fluid layers near the wall. Most

of the initial transient is spent to make the temperature field uniform in the

fluid region. All these steps are not necessary if the heat is directly provided

by an internal energy source, resulting in a faster dynamics, as can be noticed

from Figure 4.6. The CFD approach also allows simulating the triggering of

natural circulation, in contrast to the O-O model, so the dynamics is further

delayed if compared with that of the 1D numerical model. As for the different

values of the mass flow rate, in the IHG case part of the energy of the fluid is

absorbed by the walls, resulting in a lower temperature field and,

consequently, in a lower mass flow rate. However, the difference between the

two heating models is acceptable. From the comparison between the solutions

obtained with the IHG, it is also possible to notice an opposite direction of the

flow between the CFD simulation and the O-O ones. This is due to the perfect

symmetry of the system with respect to the vertical plane (see Figure 4.1),

and consequently to the lack of a preferential direction of the flow.

In Figures 4.8÷4.11, the radial distributions of the steady-state temperature

and velocity fields are shown for both the A-EHF and the IHG cases.

(a) (b)

Figure 4.8: Steady-state radial temperature distribution taken at the cooler section

(a) and at the bottom leg (b) in case of A-EHF and 10 𝑘𝑊.

(a) (b)

Figure 4.9: Steady-state radial velocity distribution taken at the cooler section (a)

and at the bottom leg (b) in case of A-EHF and 10 𝑘𝑊.



110

(a) (b)


(a) and at the bottom leg (b) in case of IHG and 10 𝑘𝑊.

(a) (b)


and at the bottom leg (b) in case of IHG and 10 𝑘𝑊.

The slight differences observed in this case are linked to the slower dynamics

of the A-EHF, for which a uniform temperature distribution is not reached

yet.

Figures 4.12 and 4.13 illustrate the results for the case at 2 𝑘𝑊. An unstable

dynamics is observed, in accordance with the prediction of the stability map,

but a slight difference is noticed from the comparison between the transient

behaviours obtained with the two numerical approaches. While unidirectional

flow oscillations are predicted with the O-O model, the 3D CFD simulations

show mass flow reversal both the A-EHF and the IHG situations.

This discrepancy can be due to the different approach adopted in modelling

the heat exchange and the pressure drop. In the O-O model, empirical

correlations are adopted, while the CFD approach allows avoiding the use of

such correlations. However, the absolute maximum values of the oscillations

are comparable for the two numerical approaches. 3D effects at the first time-

steps contributed to complicate the dynamic behaviour, as illustrated by the

radial temperature and velocity distributions reported in Figure 4.14. In

particular, flow reversal can be observed also along the vertical leg of the

loop.

4.5 Results

111

Figure 4.12: Mass flow rate transient behaviour obtained with the OpenFOAM® code for the IHG and A-EHF cases, at 2 𝑘𝑊.

(a)

(b)


2 𝑘𝑊.

The differences observed between the A-EHF and IHG situations at 10 𝑘𝑊

are recognized also in this case.

The radial distribution of the temperature and velocity fields are shown in

Figures 4.15÷4.18 for a time-step corresponding to a maximum of the mass

flow rate, for both the heating modes. A very good agreement can be observed

between the A-EHF and IHG cases, both qualitatively and quantitatively.



112

(a) (b)

Figure 4.14: Radial distribution of temperature (a) and velocity (b) taken at the right

vertical leg at 500 𝑠.

(a) (b)

Figure 4.15: Radial temperature distribution taken at the cooler section (a) and at

the bottom leg (b) in case of A-EHF, at 1430 𝑠 (maximum of the oscillation) and 2 𝑘𝑊.

(a) (b)

Figure 4.16: Radial velocity distribution taken at the cooler section (a) and at the bottom leg (b) in case of A-EHF, at 1430 𝑠 (maximum of the oscillation) and 2 𝑘𝑊.

The solutions achieved with a thermal power of 0.5 𝑘𝑊 are reported in

Figures 4.18 and 4.19. For this operative transient, the asymptotic

equilibrium reached with the CFD model differs from that attained by the O-

O model. From the information reported in the stability map, an unstable

equilibrium is expected, as confirmed also by the 1D O-O simulation, but with

the CFD approach a stable equilibrium is observed.

4.5 Results

113

(a) (b)

Figure 4.17: Radial temperature distribution taken at the cooler section (a) and at the bottom leg (b) in case of IHG, at 1660 𝑠 (maximum of the oscillation) and 2 𝑘𝑊.

(a) (b)

Figure 4.18: Radial velocity distribution taken at the cooler section (a) and at the bottom leg (b) in case of IHG, at 1660 𝑠 (maximum of the oscillation) and 2 𝑘𝑊.

Since the operative transient at 0.5 𝑘𝑊 is in proximity of the transition curve

between stable and unstable regions (see Figure 4.5), the above mentioned

effects on the heat exchange and friction loss, in addition to 3D effects, this

time lead to a different dynamic behaviour.

Figure 4.18: Mass flow rate transient behaviour obtained with the OpenFOAM® code for the IHG and A-EHF cases, at 0.5 𝑘𝑊.



114

(a)

(b)


0.5 𝑘𝑊.

A further proof is given by the results obtained with the 1D O-O model for a

power level of 0.4 𝑘𝑊 and reported in Figure 4.20. By a slight reduction of the

thermal power provided, a stable asymptotic equilibrium is observed.

Figure 4.20: Mass flow rate transient behaviour obtained with the 1D O-O model for the IHG and A-EHF cases, at 0.4 𝑘𝑊.

4.5 Results

115

As for the difference between the two heating modes for the CFD simulations,

the time required for reaching the steady-state value of the A-EHF mode is

again greater than the IHG mode, but the steady-state value achieved is

almost the same in both modes. Also in this case, the steady-state radial

temperature and velocity distributions are reported in Figures 4.21÷4.24,

showing an almost equal (qualitative and quantitative) distribution of the

flow variables both for the A-EHF and the IHG situations.

(a) (b)


(a) and at the bottom leg (b) in case of A-EHF and 0.5 𝑘𝑊.

(a) (b)


and at the bottom leg (b) in case of A-EHF and 0.5 𝑘𝑊.

(a) (b)


(a) and at the bottom leg (b) in case of IHG and 0.5 𝑘𝑊.



116

(a) (b)


and at the bottom leg (b) in case of IHG and 0.5 𝑘𝑊.

As final remark, in Table 4.4 all the results are summarized, including the

asymptotic behaviour achieved by means of both the numerical approaches

for the operative transients in exam, and the mass flow rates (steady-state

values or maximum of the oscillation mode) as well. RE between the mass

flow rate are also reported for the two heating modes, with respect to the IHG

results, and between the CFD and the O-O models, taking as reference the

latter one. RE between the A-EHF and IHG situations obtained with the CFD

model remain below ~20 %, with higher absolute values of the mass flow rate

corresponding to the A-EHF case for both the CFD and the O-O approaches.

A good agreement is observed between the results obtained with the

numerical models for the case at 10 𝑘𝑊, while a quantitative comparison of

the mass flow rate values is meaningless for the case at 0.5 𝑘𝑊. All these

differences are due to the different modelling of heat transfer and pressure

drops, to 3D effects, and to the numerical errors introduced by the different

discretization methods adopted by the two models as well.

4.6 Final remarks

In this chapter, the effect of IHG on the dynamics of a single-phase natural

circulation loop is investigated by means of the OpenFOAM® code,

considering the DYNASTY facility loop, actually under construction at the

Politecnico di Milano, and the Hitec® molten salt as working fluid.

From the consideration reported in chapter 3, the effect of thermal inertia is

taken into account, being essential for the correct prediction of the dynamic

behaviour of the fluid. The solutions of the 3D CFD approach are compared

with the solution achieved with those of the O-O model used for the design of

the facility.

4.6 Final remarks

117

Table 4.4: Summary of the results obtained for three different transients of the DYNASTY facility loop

OpenFOAM® model

𝟏𝟎 𝒌𝑾 𝟐 𝒌𝑾 𝟎. 𝟓 𝒌𝑾 Asymptotic equilibrium

A-EHF Stable Unstable (flow reversal) Stable

Asymptotic equilibrium IHG

Stable Unstable (flow reversal) Stable

Mass Flow Rate (steady state value or maximum value) A-EHF (𝒌𝒈 𝒔−𝟏)

-0.3378 0.3302 0.0651

Mass Flow Rate (steady state value or maximum

value) IHG (𝒌𝒈 𝒔−𝟏) -0.2802 0.2733 -0.0624

RE 20.557 % 20.082 % 4.327 %

1D O-O model

𝟏𝟎 𝒌𝑾 𝟐 𝒌𝑾 𝟎. 𝟓 𝒌𝑾 Asymptotic equilibrium

A-EHF Stable

Unstable (unidirectional oscillation)

Unstable (flow reversal)

Asymptotic equilibrium IHG

Stable Unstable (unidirectional

oscillation) Unstable (flow

reversal) Mass Flow Rate (steady state value or maximum value) A-EHF (𝒌𝒈 𝒔−𝟏)

-0.3297 0.3054 0.2819

Mass Flow Rate (steady state value or maximum

value) IHG (𝒌𝒈 𝒔−𝟏) -0.3029 0.2468 -0.2536

RE 8.848 % 16.582 % 11.593 %

OpenFOAM® model vs. O-O model

𝟏𝟎 𝒌𝑾 𝟐 𝒌𝑾 𝟎. 𝟓 𝒌𝑾

RE A-EHF 2.457 % 8.120 % -

RE IHG 7.494 % 10.737 % -

In particular, a different heating mode is also considered for simulating the

internal energy source in the fluid, namely an all-external heat flux along the

entire loop (except for the cooler section), which is also considered in Cammi

et al. (2016). From the obtained results, it can be noticed that the two

different heating modes induce similar effects, at least for loop characterized

by a length-to-diameter ratio very high. This information is useful, from an

experimental point of view, since it allows avoiding technical issues linked to

the experimental realization of an internal energy source.

Three different power levels are chosen to run the simulations, namely

10 𝑘𝑊, 2 𝑘𝑊 and 0.5 𝑘𝑊. At higher power, a very good agreement is observed

between the two numerical approaches. Some difference is found for the

transient at 2 𝑘𝑊 for what concern the time-dependent behaviour of the mass

flow rate, since the 1D O-O model predicts unidirectional oscillations, while

the CFD simulations show inversion in the direction of the flow. For the third



118

case, differences are also observed in the asymptotic behaviour. An unstable

equilibrium is predicted by both the 1D O-O model and the stability map,

while the CFD approach shows a stable equilibrium. This difference can be

explained by noting that the transient at 0.5 𝑘𝑊 is in proximity of the

transition curve between stable and unstable equilibria, so the different

modelling of heat transfer and pressure drops, in addition to 3D effects, can

contribute to shift the operative point from a region to another. An indirect

proof is given by the time-dependent behaviour obtained with the 1D O-O

model, for the transient at 0.4 𝑘𝑊. Further efforts to validate the developed

numerical models will be carried out once the DYNASTY facility will be

completed, providing the necessary experimental data.

References

ANSYS, 2016, http://www.ansys.com/it.

Boerema, N., Morrison, G., Taylor, R. & Rosengarten, G., 2012. Liquid sodium

versus Hitec as a heat transfer fluid in solar thermal central receiver

systems. Solar Energy, 86(9), pp. 2293-2305.




Fiorina, C. et al., 2014. Thermal-hydraulics of internally heated molten salts

and application to the Molten Salt Fast Reactor. Journal of Physics, 501(1),

pp. 1-10.

GIF, 2014. Generation IV International Forum Annual Report,

http://www.gen4.org/.

Luzzi, L., Cammi, A., DiMarcello, V. & Fiorina, C., 2010. An approach for the

modelling and the analysis of the MSR thermo-hydrodynamic behaviour.

Chemical Engineering Science, 65(16), pp. 4873-4883.

Olivares, R., 2012. The thermal stability of molten nitrite/nitrates salt for

solar thermal energy storage in different atmospheres. Solar Energy, 86(9),

pp. 2576-2583.




http://www.ansys.com/it

http://www.gen4.org/



References

119

with internally heated fluids. Chemical Engineering Science, Volume 145, pp.

108-125.

Pini, A., Cammi, A., Luzzi, L. & Ruiz, D. E., 2014. Linear and nonlinear

analysis of the dynamic behaviour of natural circulation with internally

heated fluid. Proceedings of 10th International Topical Meeting on Nuclear

Therma-Hydraulic, Okinawa, Japan.



Chemical Engineering Science, Volume 126, pp. 573-583.

SAMOFAR, 2015, http://samofar.eu/.

Serp, J. et al., 2016. The Molten Salt Reactor (MSR) in generation IV:

Overview and perspective. Progress in Nuclear Energy, Volume 77, pp. 308-

319.

Serrano-Lopez, R., Fradera, J. & Cuesta-Lopez, S., 2013. Molten salt database

for energy applications. Chemical Engineering and Processing, Volume 73,

pp. 87-102.

Srivastava, A. et al., 2015. Experimental and theoretical studies on the

natural circulation behaviour of molten salt loop. Applied Thermal Energy,

Volume 98, pp. 513-521.

http://samofar.eu/

121

CONCLUSION

In this thesis work, natural circulation phenomena are numerically studied

by adopting a CFD approach. In particular, standard solvers of the

OpenFOAM® finite volume library are modified in order to investigate also

the case of internally heated fluids, which recently have gained more

attention both from a scientific and a practical point of view, in light of the

renewed interest for new generation nuclear power plants, whose main

example is given by the Molten Salt Reactor.

The modified solvers are firstly assessed adopting a simple case, namely the

differentially-heated square cavity, obtaining a very good agreement with

several benchmarks available in literature for a wide range of Rayleigh

number. Through a sensitivity study on different RANS turbulence models

adopted in the case of turbulent flow regime, it is observed that the better

results are obtained with standard 𝑘 − 휀 model with scalable wall functions

and with the Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model without wall functions. An extensive

study is also carried out for both external and internal heating, showing the

main effects induced by the internal heat generation on flow pattern and

dynamic behaviour, considering Rayleigh numbers in the range 106 ÷ 1.58 ×

109 and four different values of the 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ ratio. The general trend is the

symmetry breaking of the flow pattern when the internal heat generation is

predominant with respect to the external heating. Another important

consequence is the inversion of the direction of the heat flux when the effect

of internal heat generation is greater than the external heating. Also the

Prandtl number of the fluid affects the flow pattern and the transient-

behaviour, as it is observed comparing the results obtained for 𝑃𝑟 = 0.71 (air)

with those obtained for 𝑃𝑟 = 0.0321 (lithium-beryllium alloy).

In the second part of the thesis, the proposed models are validated for the

case of a single-phase natural circulation loop, a configuration widely adopted

for the design of systems based on free convection. The validation is carried

out by using the experimental data provided by the L2 facility of the DIME-

TEC Labs (Genova University) and available from IAEA. Three different

models are developed, investigating both the effect of thermal inertia and

heat exchanger modelling. The results confirm that the presence of piping

materials are necessary to correctly predict the dynamics of natural

circulation loops, as reported in Cammi et al. (2016). Moreover, the complete

coaxial cylinder heat exchanger allows obtaining a better agreement for the

absolute values of the fluid temperature field. For this last model, two

different turbulence models are also considered (namely, Menter 𝑆𝑆𝑇 𝑘 − 𝜔

CONCLUSION

122

and Realizable 𝑘 − 휀), both with scalable wall functions, in order to show how

the results are affected by this choice. The best results are those obtained

with the 𝑆𝑆𝑇 𝑘 − 𝜔. At last, a comparison with a 1D O-O model is also

reported, showing an overall good agreement between the numerical models

and with the experimental data as well.

In the last part of this work, the case of a loop with internal energy sources is

taken into account, simulating the dynamic behaviour of the DYNASTY

facility, which is going to be realised at Energy Labs of the Politecnico di

Milano. The results obtained with the 3D CFD approach are compared with

the 1D O-O model previously mentioned and used for the design of the facility

itself. In addition to the effect of internal energy sources, an alternative

heating mode is also considered by providing an imposed external heat flux

along the entire loop, except for the cooler section. As reported in Cammi et

al. (2016), this strategy allows reproducing the effect of the internal heat

generation, avoiding the non-trivial technical issues linked to the induction of

the distributed volumetric source inside the loop, from an experimental point

of view. CFD simulations run in this work confirm this prediction, showing

only slight differences between these two heating modes.

From the comparison of the results obtained with the different numerical

models, some differences are observed in the region near the transition curve

on the stability map, but these differences mainly derive from 3D effects and

the different approach adopted in modelling the heat exchange and the

pressure loss in the two models.

Purpose of future works will be to improve the obtained results by considering

advanced turbulence models, such as RANS Reynolds Stress Model (seven-

equation turbulence model) or LES turbulence models. In addition, the

results obtained for the DYNASTY loop in this thesis will be also validated

using the experimental data provided by the facility, once it will be

completed.

125

APPENDIX: Assessment of the OpenFOAM® for the

case of 2D conventional natural circulation in a

differentially heated air-filled square cavity at

different Rayleigh numbers

A.1 Introduction

The development of more and more reliable CFD codes is based on their

validation with experimental data. One of the most common experimental

benchmark is referred to the buoyant cavity, widely chosen thanks to the

relatively simple configuration and the great number of works on this topic.

In spite of geometrical simplicity, a full description of occurring phenomena

has not been obtained yet. Moreover, even though many data are available in

literature about low Rayleigh number regimes, less studies have been

conducted on transition and turbulent regimes, because of their intrinsic

unsteadiness and few accuracy of the turbulence models used in numerical

simulations.

Purpose of this Appendix is mainly to familiarize with the pre- and post-

processing procedure of the OpenFOAM® code, learning to set a well-known

case such that of the buoyancy-driven cavity.

In the first part, the results obtained in a wide range of Rayleigh number are

compared with different benchmarks available in literature. In particular, the

results are organized into three parts: laminar flow regime (𝑅𝑎 = 103 ÷ 106),

transition flow regime (𝑅𝑎 = 107 ÷ 108) and turbulent flow regime (𝑅𝑎 =

1.58 × 109).

In the second part of the Appendix, wall materials of the cavity are taken into

account. In this way, a conjugate heat transfer solver of the OpenFOAM® code

is also presented. In general, the presence of TI can affect the dynamics of

natural circulation systems (e.g., natural circulation loops, as shown in

chapters 3 and 4). For the considered square cavity, in which external

temperatures are imposed, no effects are expected. The simulations are run

considering only the cases at 𝑅𝑎 = 106 (for the laminar flow regime) and

𝑅𝑎 = 1.58 × 109 (for the turbulent flow regime). Three different materials are

chosen for TI, namely: AISI-316, pure Aluminium and pure Copper.

APPENDIX: Assessment of the OpenFOAM® for the case of 2D conventional natural

circulation in a differentially heated air-filled square cavity at different Rayleigh

numbers

126

A.2 Modelling and implementation in OpenFOAM®

A sketch of the system is depicted in Figure A.1, while in Table A.1 the

thermal IC and BC, the geometrical size (𝐿) and depth (𝑊) of the cavity and

the values of the thermal expansion coefficient (𝛽), of the momentum

diffusivity (𝜈) and of the Prandtl number (𝑃𝑟 ) at the reference temperature

are summarized for all the cases.

Although the system is a 2D cavity, OpenFOAM® runs only with 3D

configurations, so a depth must be considered (see Table A.1).

The code is run using the buoyantSourceBoussinesqSimpleFoam solver in the

range of Rayleigh number 103 ÷ 107, where a steady-state is still reached, and

buoyantSourceBoussinesqPimpleFoam solver for 𝑅𝑎 = 108 and 1.58 × 109,

using the SIMPLE and PIMPLE algorithm for the pressure-velocity coupling,

respectively. Meshes are generated with the blockMesh utility of the

OpenFOAM® code and several uniform grids are used in order to reach a grid-

independent solution. Thermal properties values of the fluid are reported in

the transportConstant file.

Figure A.1: Differentially heated 2D cavity.

Table A.1: Thermal IC and BC, size of the cavity, thermal properties at 300 K (Bergman et al., 2011)

Laminar and transition flow regimes


300.5 301 300 1 0.1 3 × 10−3 10−5 0.71

Turbulent flow regimes


300 320 280 0.75 1.5 3.4 × 10−3 1.59 × 10−5 0.71

A.3 Results

127

The different Rayleigh numbers are obtained simply changing the value of

the gravitation acceleration, applying the definition of the dimensionless

number, as follows:

𝑅𝑎 = 𝐺𝑟𝑃𝑟 =𝑔𝛽𝐿3Δ𝑇𝑃𝑟

𝜈2 ⇒ 𝑔 =

𝑅𝑎 𝜈2

𝛽𝐿3Δ𝑇𝑃𝑟 (𝐴. 1)

The flow regime is specified in the turbulenceProperties file: laminar is set for


regime. The particular turbulence model used is specified in the

RASProperties file.

The tolerance is set to 10−7 for all the physical variables, and the following

relaxation factors are imposed: 0.3 for pressure, 0.7 for velocity and 1 for

temperature.

Different numerical schemes are used in running the code. Bounded Gauss

linear discretization is used as scheme for the divergence terms for laminar

and transition flow regimes while upwind discretization is employed for

turbulent flow regime. The default discretization schemes are used for the

other terms for all flow regimes. Even though upwind scheme is characterized

by numerical diffusion and a first-order accuracy, while Gauss linear is a

second-order scheme, its simplicity allows obtaining a more stable numerical

solution where a steady-state no longer exists and oscillations are present, as

the case of turbulent flows.

At last, the maximum Courant number is set at 0.1 for the SIMPLE

algorithm, in order to obtain more accurate solution, while for the PIMPLE

algorithm it is sufficient a Courant number equals to 5.

A.3 Results

A.3.1 Laminar flow regime (𝑅𝑎 = 103 ÷ 106)

The results in this range of Rayleigh numbers are compared with the

numerical solutions given by Corzo et al. (2011) and De Vahl Davis (1983).

The quantities reported are:

the maximum dimensionless x-velocity component at the mid-width with

its location;

the maximum dimensionless y-velocity component at the mid-height with

its location;

the maximum and minimum local Nusselt number at 𝑥 = 0 with the

corresponding locations.



numbers

128

The dimensionless velocities and the dimensionless coordinates are defined

as:

𝑈 =𝑢𝐿

𝛼=𝑢𝐿𝑃𝑟

𝜈, 𝑉 =

𝑣𝐿

𝛼=𝑣𝐿𝑃𝑟

𝜈, 𝑋 =

𝑥

𝐿, 𝑌 =

𝑦

𝐿 (𝐴. 2)

The local Nusselt number is calculated using its definition, the Newton’s law

of cooling and the Fourier’s law applied to the fluid at the wall:

𝑁𝑢 =ℎ𝐿

𝜆, 𝑞′′ = ℎΔ𝑇, 𝑞′′ = −𝜆

𝜕𝑇

𝜕𝑥|𝑤𝑎𝑙𝑙

⇒ 𝑁𝑢 = −𝐿

Δ𝑇

𝜕𝑇

𝜕𝑥|𝑤

(𝐴. 3)

In Tables A.2÷A.5, the grid-independence study is reported for the velocity

components at different Rayleigh numbers and the RE between two

consecutive grid results is shown. Also extra-digits are quoted in order to

highlight the difference in the results obtained with the several grids

considered. In Table A.6, the comparison between the results obtained with

the finer meshes and the benchmarks are summarized with the RE. Figures

of temperature and velocity distributions are reported after each table.

Table A.2: Grid Sensitivity 𝑅𝑎 = 103

(𝟐𝟎 × 𝟐𝟎) (𝟒𝟎 × 𝟒𝟎) (𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟔𝟎 × 𝟏𝟔𝟎)

𝑼 3.58983 3.62688 3.64388 3.64656 3.64634 3.64686

RE / 1.03 % 0.47 % 0.074 % 0.006 % 0.01 %

𝒚 0.8 0.82 0.81 0.81 0.81 0.81

RE / 2.5 % 1.22 % 0 0 0

𝑽 3.61303 3.67683 3.69102 3.69491 3.69424 3.69544

RE / 1.76 % 0.39 % 0.1 % 0.02 % 0.03 %

𝒙 0.2 0.18 0.18 0.18 0.18 0.18

RE / 10 % 0 0 0 0

Figure A.2: Temperature and velocity distribution for 𝑅𝑎 = 103.

A.3 Results

129


(𝟐𝟎 × 𝟐𝟎) (𝟒𝟎 × 𝟒𝟎) (𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟔𝟎 × 𝟏𝟔𝟎)

𝑼 15.71770 16.06993 16.14739 16.16601 16.16478 16.17167

RE - 2.24 % 0.48 % 0.12 % 0.013 % 0.042 %

𝒚 0.8 0.82 0.82 0.82 0.82 0.82

RE - 2.5 % 0 0 0 0

𝑽 18.75216 19.39045 19.55787 19.60345 19.59628 19.61205

RE - 3.4 % 0.86 % 0.23 % 0.036 % 0.08 %

𝒙 0.1 0.12 0.12 0.12 0.12 0.12

RE - 20 % 0 0 0 0



(𝟐𝟎 × 𝟐𝟎) (𝟒𝟎 × 𝟒𝟎) (𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟔𝟎 × 𝟏𝟔𝟎)

𝑼 34.16314 34.63884 34.69507 34.70075 34.70373 34.70622

RE - 1.39 % 0.16 % 0.016 % 0.009 % 0.007 %

𝒚 0.85 0.85 0.85 0.85 0.85 0.85

RE - 0 0 0 0 0

𝑽 61.154004 65.98570 67.73506 68.22837 68.14189 68.31095

RE - 7.9 % 2.65 % 0.73 % 0.13 % 0.25 %

𝒙 0.05 0.07 0.07 0.07 0.07 0.07

RE - 40 % 0 0 0 0




numbers

130


(𝟒𝟎 × 𝟒𝟎) (𝟖𝟎 × 𝟖𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟔𝟎 × 𝟏𝟔𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎)

𝑼 66.18166 65.05105 64.98105 64.93880 64.89414 64.87263

RE - 1.71 % 0.11 % 0.065 % 0.069 % 0.033 %

𝒚 0.85 0.85 0.85 0.85 0.85 0.85

RE - 0 0 0 0 0

𝑽 124.53858 214.94682 218.12194 217.58731 218.42511 219.57105

RE - 10.49 % 1.48 % 0.25 % 0.38 % 0.52 %

𝒙 0.05 0.04 0.04 0.04 0.04 0.04

RE - 20 % 0 0 0 0


Assuming that a solution is grid-independent if the relative error between

two successive solutions is of the order of 1 %, it can be noticed from the

results in Tables A.2÷A.5 that the grid-independent solution is already

reached with a grid (100 × 100).

Initially, the QUICK discretization was used as scheme for the divergence

terms, but convergence problems with the SIMPLE algorithm were found.

However, the results obtained with the linear scheme do not differ

significantly from those obtained with the QUICK scheme, as shown from the

comparison with Corzo et al. (2011) reported in Table A.6. Only the grid

(40 × 40) for the case at 𝑅𝑎 = 106 gives problem in the accuracy of the results

with the linear scheme, therefore the solution achieved by means of the

QUICK scheme (with a number of iteration of 10000) is reported.

As can be noticed in Table A.6, an good agreement is found between the data

obtained with the OpenFOAM® code used in this work and the results

available in literature, as could be expected form the regular flow regime

typical of low Rayleigh numbers. Numerically, 𝜕𝑇 𝜕𝑥⁄ |𝑤 for the evaluation of

the local Nusselt number is computed with a two points finite difference

between the first two points in the temperature distribution along x-axis. The

greatest errors obtained are those of the locations of the maximum local

Nusselt and the values of the minimum local Nusselt. For example, the

A.3 Results

131

location of the maximum local Nusselt for 𝑅𝑎 = 103 in Corzo et al. (2011) is

very different from that obtained in the present work and the RE with the

value given in De Vahl Davis (1983) is 14.94 %. In the case at 𝑅𝑎 = 104, on the

contrary, the greatest RE is with the data of De Vahl Davis (1983), equal to

53.02 %, while for Corzo et al. (2011) is 17.65 %.

Table A.6: Comparison with benchmarks

𝑹𝒂 = 𝟏𝟎𝟑 𝑹𝒂 = 𝟏𝟎𝟒

This work Corzo et al.

(2011) De Vahl

Davis (1983) This work

Corzo et al. (2011)

De Vahl Davis (1983)

𝑼 3.647 3.640 3.634 16.172 16.281 16.182

RE / 0.19 % 0.35 % / 0.67 % 0.064 %

𝒚 0.81 0.812 0.813 0.82 0.822 0.823

RE / 0.25 % 0.37 % / 0.24 % 0.36 %

𝑽 3.695 3.700 3.679 19.612 19.547 19.509

RE / 0.12 % 0.45 % / 0.33 % 0.53 %

𝒙 0.18 0.177 0.179 0.12 0.123 0.120

RE / 1.69 % 0.56 % / 2.24 % 0.0 %

𝑵𝒖𝒎𝒂𝒙 1.5 1.505 1.501 3.5 3.538 3.545

RE / 0.33 % 0.067 % / 1.07 % 1.27 %

𝒚𝒎𝒂𝒙 0.1 0.001 0.087 0.07 0.085 0.149

RE / - 14.94 % / 17.65 % 53.02 %

𝑵𝒖𝒎𝒊𝒏 0.7 0.691 0.694 0.6 0.691 0.592

RE / 1.3 % 0.86 % / 13.17 % 1.35 %

𝒚𝒎𝒊𝒏 0.99 1 1 0.99 1 1

RE / 1 % 1 % / 1 % 1 %

𝑹𝒂 = 𝟏𝟎𝟓 𝑹𝒂 = 𝟏𝟎𝟔


(2011) De Vahl

Davis (1983) This work

Corzo et al. (2011)

De Vahl Davis (1983)

𝑼 34.706 34.928 34.810 64.873 64.558 65.33

RE / 0.63 % 0.3 % / 0.49 % 0.7 %

𝒚 0.85 0.859 0.855 0.85 0.851 0.851

RE / 1.05 % 0.58 % / 0.12 % 0.12 %

𝑽 68.311 68.878 68.220 219.571 221.57 216.75

RE / 0.82 % 0.13 % / 0.9 % 1.3 %

𝒙 0.07 0.067 0.066 0.04 0.067 0.0387

RE / 4.48 % 6.06 % / 40.3 % 3.36 %

𝑵𝒖𝒎𝒂𝒙 7.8 7.765 7.761 17.6 17.708 18.076

RE / 0.45 % 0.5 % / 0.61 % 2.63 %

𝒚𝒎𝒂𝒙 0.089 0.080 0.085 0.04 0.0404 0.0456

RE / 11.25 % 4.7 % / 0.99 % 12.28 %

𝑵𝒖𝒎𝒊𝒏 0.7 0.726 0.736 0.9 0.977 1.005

RE / 3.58 % 4.89 % / 7.88 % 10.45 %

𝒚𝒎𝒊𝒏 0.99 1 1 0.99 0.998 1

RE / 1 % 1 % / 0.8 % 1 %



numbers

132

Nonetheless, these errors are consistent with those founded in literature, as

shown by the comparison between the two benchmarks, and for this reason

they can be considered acceptable.

A.3.2 Transition flow regime (𝑅𝑎 = 107 ÷ 108)

In the range of Rayleigh 107 ÷ 108, the configuration of the flow starts to

become unsteady. This is due to the transition from laminar to turbulent flow

regime, whose limit Rayleigh number is found to be ~2 × 108, as reported by

Le Quéré (1991). The case at 𝑅𝑎 = 107 still reaches convergence with the

steady-state algorithm, while for the case at 𝑅𝑎 = 108 it is necessary to adopt

the unsteady PIMPLE algorithm (the simulation is conducted for 2500 𝑠𝑒𝑐).

Simulations require finer grids and, consequently, a higher computational

cost because of the stronger coupling between the y-momentum equation and

the energy equation due to the unsteady behaviour of the flow.

In Tables A.7÷A.8, the results with different grids are reported and the RE

between two consecutive grid solutions is shown, considering also the extra-

digits for the comparison. Table A.9 shows the comparison with the solutions

given by Corzo et al. (2011) and by Le Quéré, (1991) with the RE Figures of

temperature and velocity distributions are reported after each table.


(𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑼 148.6733 148.7521 148.6541 148.7166

RE - 0.053 % 0.066 % 0.042 %

𝒚 0.88 0.88 0.88 0.879

RE - 0 0 0.11 %

𝑽 684.1148 693.5756 688.7767 695.7538

RE - 1.38 % 0.69 % 1.01 %

𝒙 0.02 0.02 0.02 0.021

RE - 0 0 5 %


A.3 Results

133


(𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑼 329.2582 330.3261 330.9814

RE - 0.32 % 0.2 %

𝒚 0.93 0.93 0.929

RE - 0 0.11 %

𝑽 2143.36 2110.84 2173.71

RE - 1.52 % 2.98 %

𝒙 0.01 0.01 0.012

RE - 0 20 %


Table A.9: Comparison with benchmarks

𝑹𝒂 = 𝟏𝟎𝟕 𝑹𝒂 = 𝟏𝟎𝟖


(2011) Le Quéré

(1991) This work

Corzo et al. (2011)

Le Quéré

(1991)

𝑼 148.717 145.84 148.58 330.981 299.156 321.876

RE / 1.97 % 0.09 % / 10.64 % 2.83 %

𝒚 0.879 0.884 0.879 0.929 0.921 0.928

RE / 0.56 % 0.0 % / 0.87 % 0.11 %

𝑽 695.754 704.094 699.236 2173.71 2233.35 2222.39

RE / 1.18 % 0.5 % / 2.67 % 2.19 %

𝒙 0.021 0.0217 0.021 0.012 0.012 0.012

RE / 3.23 % 0.0 % / 0.0 % 0.0 %

𝑵𝒖𝒎𝒂𝒙 39.96 40.594 39.39 93.44 90.294 87.24

RE / 1.56 % 1.45 % / 3.48 % 7.11 %

𝒚𝒎𝒂𝒙 0.018 0.017 0.018 0.008 0.008 0.008

RE / 5.88 % 0.0 % / 0.0 % 0.0 %

𝑵𝒖𝒎𝒊𝒏 1.41 1.365 1.366 2.3 1.9061 1.919

RE / 3.3 % 3.22 % / 20.67 % 20.42 %

𝒚𝒎𝒊𝒏 0.99 0.998 1 0.99 0.999 1

RE / 0.8 % 1 % / 0.9 % 1 %

Good results are obtained using OpenFOAM® even for these Rayleigh

numbers. In particular, for 𝑅𝑎 = 107 the RE are still very low, especially with

the results of Le Quéré (1991), while for 𝑅𝑎 = 108 the RE are slightly greater



numbers

134

than the previous cases. This can be explained by the different numerical

methods used in the several studies: in Corzo et al. (2011) the SIMPLE

algorithm was applied also in this last case and the QUICK discretization

was used; Le Quéré (1991) used a pseudo-spectral method, combining spatial

expansion in series of Chebyshev polynomial with a finite-difference time-

stepping scheme (see Le Quéré (1991) for more detail).

The less accurate solutions are the values of minimum and maximum Nusselt

number, linked to the suddenly changes of the velocity and temperature

distributions in the boundary layer in the transition flow regime. This time,

the factor 𝜕𝑇 𝜕𝑥⁄ |𝑤 for the evaluation of the local Nusselt number is computed

interpolating the first ten points of the numerical solution of the temperature

along the x-axis with a third grade polynomial and calculating the first

derivative for 𝑥 = 0, since inaccurate values would be obtained with the two

points finite difference.

A.3.3 Turbulent flow regime (𝑅𝑎 = 1.58 × 109)

The results are compared with the experimental benchmark given by Ampofo

& Karayiannis (2003). In general, experiments for the study of turbulence

flow regime present several problems.

As reported by Tian & Karayiannis (2000), the main problems in the

experimental set-up are: the adiabatic conditions on the top and bottom walls

of the cavity, the 2D approximation and the measurement of the quantities of

interest as accurately as possible without perturbing the system. In an

experiment is not possible to realize a perfect adiabatic boundary condition

and this fact can represent a possible source of discrepancy in the results

obtained with numerical simulations. The 2D approximation should be valid

if the length in the third direction is long enough so that its influence can be

neglected. This condition can be achieved with a horizontal aspect ratio (𝐴𝑅𝑧)

greater than 1.8. In this study is set 𝐴𝑅𝑧 = 2. About problems in the

measurement see Tian & Karayiannis (2000) and Ampofo & Karayiannis

(2003).

Only the maximum of the dimensionless v-component of the velocity, with is

location, and the local Nusselt number at the hot wall are considered at the

mid-height of the cavity. While the Nusselt number is defined as in the

previous sections, a different definition is used for the dimensionless velocity:

𝑉 =𝑣

√𝑔𝛽𝐿Δ𝑇 (𝐴. 4)

A.3 Results

135

Average values for 𝑦+ in the horizontal and vertical walls are reported for the

near-wall treatment.

Simulations are conducted for 500 𝑠𝑒𝑐, using several RANS models in order to

assess how the choice of the turbulence model affects the results. In

particular three two-equation models are used: standard 𝑘 − 휀 model, Wilcox

𝑘 − 𝜔 model and Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model. Both HR and LR wall functions are

used for the near-wall treatment, even though the experiment consists in the

study of low turbulence in a cavity, in order to show how the near-wall

treatment can influence the numerical solution.

About IC, information on turbulent quantities is rarely available. Turbulent

kinetic energy is estimated using the definition of the turbulence intensity:

𝑘 =2

3(𝑈𝑟𝑒𝑓ℐ)

2 (𝐴. 5)

where a turbulence intensity of 5 % is considered, while 휀 and 𝜔 are

calculated using auxiliary relations reported in chapter 1, when the eddy

viscosity was defined:

휀 = 𝐶𝜇34⁄𝑘32⁄

ℓ, 𝜔 =

𝑘12⁄

ℓ, ℓ = 0.07 𝐿 (𝐴. 6)

Grid-dependence solution is studied using different uniform meshes for all

the turbulence models, as reported in Tables A.10÷A.15, and RE between two

successive grid results is reported for each case. Also in this case extra-digits

are reported in order to compare the results. In Table A.16, the solution

obtained with the finer grid is compared with the data provided in the

benchmark, reporting the RE.

Table A.10: Grid sensitivity: standard 𝑘 − 휀 model (Hi-Reynolds wall function)

(𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑽 0.20622 0.21939 0.22738 0.23357

RE - 6.386 % 3.643 % 2.722 %

𝒙 0.009 0.007 0.007 0.006

RE - 22.222 % 0 14.286 %

𝒚+ (h.w.) 1.58 1.3165 1.1263 0.97774

𝒚+ (v.w.) 5.42095 5.05554 4.71317 4.39579

𝑵𝒖𝒎𝒂𝒙 45.9336 49.3768 51.8589 53.6336

RE - 7.496 % 5.027 % 3.264 %



numbers

136

Table A.11: Grid sensitivity: standard 𝑘 − 휀 model (Low-Reynolds wall function)

(𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑽 0.19193 0.20283 0.21184 0.21845

RE - 5.678 % 4.441 % 3.121 %

𝒙 0.009 0.008 0.007 0.006

RE - 11.111 % 12.5 % 14.286 %

𝒚+ (h.w.) 1.18210 1.51529 1.26939 1.11276

𝒚+ (v.w.) 5.42257 5.12761 4.8536 4.59473

𝑵𝒖𝒎𝒂𝒙 47.6625 52.197 55.638 58.0309

RE - 9.514 % 6.592 % 4.301 %

Table A.12: Grid sensitivity: Wilcox 𝑘 − 𝜔 model (Hi-Reynolds wall function)

(𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑽 0.18882 0.20202 0.20581 0.21131

RE - 6.988 % 1.876 % 2.676 %

𝒙 0.009 0.007 0.007 0.006

RE - 22.222 % 0 14.286 %

𝒚+ (h.w.) 0.340987 0.236176 0.171932 0.13123

𝒚+ (v.w.) 3.44584 3.02664 2.6344 2.27808

𝑵𝒖𝒎𝒂𝒙 45.0382 48.1039 50.2091 51.6652

RE - 6.807 % 4.376 % 2.9 %

Table A.13: Grid sensitivity: Wilcox 𝑘 − 𝜔 model (Low-Reynolds wall function)

(𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑽 0.18884 0.20212 0.20584 0.21140

RE - 7.032 % 1.842 % 2.7 %

𝒙 0.009 0.007 0.007 0.006

RE - 22.222 % 0 14.286 %

𝒚+ (h.w.) 1.48755 1.24412 1.07438 0.945669

𝒚+ (v.w.) 6.04266 5. 26422 5.21127 4.81603

𝑵𝒖𝒎𝒂𝒙 45.042 48.1041 50.1999 51.6487

RE - 6.798 % 4.357 % 2.886 %

Table A.14: Grid sensitivity: Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model (Hi-Reynolds wall function)

(𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑽 0.20411 0.21784 0.22421 0.23331

RE - 8.698 % 2.926 % 4.057 %

𝒙 0.009 0.007 0.007 0.006

RE - 22.222 % 0 14.286 %

𝒚+ (h.w.) 0.32679 0.22924 0.16808 0.12905

𝒚+ (v.w.) 3.46419 3.04962 2.66217 1.99633

𝑵𝒖𝒎𝒂𝒙 46.0455 49.0322 51.0141 52.1143

RE - 6.486 % 4.042 % 2.157 %

A.3 Results

137

Table A.15: Grid sensitivity: Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model (Low-Reynolds wall function)

(𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟐𝟎𝟎 × 𝟐𝟎𝟎) (𝟐𝟐𝟎 × 𝟐𝟐𝟎) (𝟐𝟒𝟎 × 𝟐𝟒𝟎)

𝑽 0.20018 0.21769 0.22310 0.23320

RE - 8.78 % 2.893 % 4.113 %

𝒙 0.009 0.007 0.007 0.006

RE - 22.222 % 0 14.286 %

𝒚+ (h.w.) 1.48435 1.2377 1.06659 0.939119

𝒚+ (v.w.) 6.09825 5.66806 5.25231 4.85545

𝑵𝒖𝒎𝒂𝒙 46.0672 49.0341 51.0051 52.0742

RE - 6.44 % 4.019 % 2.096 %

Table A.16: Comparison with benchmark

𝒌 − 𝜺 𝑯𝑹 𝒌 − 𝜺 𝑳𝑹 Ampofo & Karayiannis

(2003)

𝑽 0.2336 0.2184 0.2127

RE 9.814 % 2.704 % -

𝒙 0.006 0.006 6.6667 × 10−3

RE 10 % 10 % -

𝑵𝒖𝒎𝒂𝒙 53.633 58.031 58

RE 7.67 % 0.053 % -

𝒌 −𝝎 𝑯𝑹 𝒌 − 𝝎 𝑳𝑹 Ampofo & Karayiannis

(2003)

𝑽 0.2113 0.2114 0.2127

RE 0.652 % 0.609 % -

𝒙 0.006 0.006 6.6667 × 10−3

RE 10 % 10 % -

𝑵𝒖𝒎𝒂𝒙 51.665 51.649 58

RE 10.922 % 10.95 % -

𝑺𝑺𝑻 𝒌 − 𝝎 𝑯𝑹 𝑺𝑺𝑻 𝒌 − 𝝎 𝑳𝑹 Ampofo & Karayiannis

(2003)

𝑽 0.2333 0.2332 0.2127

RE 9.69 % 9.639 % -

𝒙 0.006 0.006 6.6667 × 10−3

RE 10 % 10 % -

𝑵𝒖𝒎𝒂𝒙 52.114 52.074 58

RE 10.148 % 10.217 % -

Grid-independent solution is not reached since simulations with finer meshes

require higher computational time than those with laminar and transition

flow regimes. Therefore the solution is considered to be grid-independent

when the RE is less than 5 %. Grids (160 × 160) and (180 × 180) satisfy this

assumption.

Values obtained for 𝑦+ in vertical walls are always greater than 1, which

corresponds to the viscous-sub layer. The location of the maximum of the

velocity is the same for all the turbulence models while differences arise in

predicting the maximum of the velocity and the Nusselt number.



numbers

138

Considerations on the performance of turbulence models can be summarized

as follow:

Standard 𝑘 − 휀 model: This model is valid especially for fully turbulent

flows. For this reason it is mostly affected by the choice of the near-wall

treatment. Values of 𝑦+ is almost the same in both cases.

On one hand, the results obtained with HR wall function are quite

different than that of Ampofo & Karayiannis (2003), linked to the use of

wall-damping functions in the near-wall region, but RE remain less than

10 %. On the other hand, simulations with LR wall function show the best

agreement with the benchmark for both the maximum of the v-velocity

component and the Nusselt number, thanks to its different approach in

the near-wall treatment. This result is consistent since 𝑘 − 휀 model works

well with simple flows. Except the case of 𝑥𝑚𝑎𝑥, RE are less than 3 %,

which represents an acceptable results.

Wilcox 𝑘 − 𝜔 model: In OpenFOAM® a single wall function is available for

the specific dissipation 𝜔 for any value of the 𝑅𝑒. The main difference

between the results in Table A.12 and Table A.13 is the value 𝑦+.

However, this difference doesn’t influence the results since 𝑦+ is always

greater than 1 and less than 5. This model gives good results for the

maximum of the velocity but underestimates the Nusselt number, with a

RE of about 11 %.

Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model: Even in this case, the value 𝑦+ is the only

difference between High and Low-Reynolds wall functions. This time,

however, despite a slight improvement in the prediction of the Nusselt

number compared to the results obtained with 𝑘 − 𝜔 model, values of the

maximum of the velocity are fairly different from experimental results.

In a geometry such that of the buoyancy-driven cavity, the 𝑆𝑆𝑇 𝑘 − 𝜔

model works better without wall function, using finer mesh such that

𝑦+ ≤ 1. Moreover, this model shows better performance than standard

𝑘 − 휀 and Wilcox 𝑘 − 𝜔 models especially with more complex flows or flows

with adverse pressure gradient.

As for the first point, two other simulations are run using non uniform

grids, refined at the walls, in order to show how results change. Results

are reported in the following Table A.17 with RE.

Fixed values are used as BC of turbulent quantities. Solutions can be

considered grid-independent since the RE between two successive grids is

less than 1 % for velocity and the Nusselt number. For the latter, it can be

noticed an improvement in its prediction while the RE for the velocity is

greater than 4 %.

A.3 Results

139

Table A.17: Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model (non-uniform grid)

(𝟐𝟒𝟎 × 𝟐𝟒𝟎) (𝟑𝟏𝟎 × 𝟑𝟏𝟎) Ampofo & Karayiannis

(2003)

𝑽 0.2422 0.2421 0.2127

RE 13.825 % 13.778 % -

𝒙 0.006 0.007 6.6667 × 10−3

RE 10 % 5 % -

𝑵𝒖𝒎𝒂𝒙 58.997 59.470 58

RE 1.719 % 2.535 % -

As final remark, in Figures A.8 and A.9 the temperature and velocity

distributions are reported for standard 𝑘 − 휀 model with LR wall functions

and Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model with non-uniform grid. From velocity profiles it

can be noticed that 𝑆𝑆𝑇 𝑘 − 𝜔 model predicts a thinner boundary layer than

𝑘 − 휀 model, thanks to the grid refinement in the near-wall region. Even

though the former is qualitatively more realistic, 𝑆𝑆𝑇 𝑘 − 𝜔 model

overestimates the maximum of the velocity.

In a similar way, thermal stratification is shown more clearly in the

temperature distribution given by the 𝑆𝑆𝑇 𝑘 − 𝜔 model but, even in this case,

the Nusselt number is better predicted by 𝑘 − 휀 model.

Figure A.8: Temperature and velocity distribution for 𝑅𝑎 = 1.58 × 109 with Low-Reynolds 𝑘 − 휀 model.

Figure A.9: Temperature and velocity distribution for 𝑅𝑎 = 1.58 × 109 with the 𝑆𝑆𝑇 𝑘 − 𝜔 model.



numbers

140

A.4 Effects of Thermal Inertia

So far, the presence of the TI of the cavity has been neglected. In the second

part of this Appendix, two of the cases previously analysed are taken into

account, namely the cases at 𝑅𝑎 = 106 (for the laminar flow regime) and

𝑅𝑎 = 1.58 × 109 (for turbulent flow regime), respectively. The primary aim is

to show how conjugate heat transfer can be treated by means of OpenFOAM®.

For the case of a buoyancy-driven cavity with an imposed external

temperature difference, no influence of TI is expected on the dynamic

behaviour of the fluid.

A.4.1 Modelling and implementation in OpenFOAM®

The dimensions of the cavity in the two cases are the same as those used in

the first part of this work, and reported in Table A.1. TI are introduced to the

left and right sides, with the same height and a width equal to an half of that

of the cavity.

The modified solver chtSourceMultiRegionFoam is adopted. This solver surely

gives more accurate solution, but it requires high computational cost, even for

simple cases as that of the laminar flow regime. Even in this case, meshes are

generated with the blockMesh utility of the OpenFOAM® code, and several

grids are used in order to reach a grid-independent solution. In particular,

two grids are considered for the two cases: uniform grids (100 × 100) and

(120 × 120) for the laminar flow regime, uniform grids (160 × 160) and

(180 × 180) and non-uniform grid (310 × 310) for the turbulent flow regime,

taking as reference the results obtained without TI, in the first part of the

Appendix. The regionProperties file allows distinguishing solid regions from

fluid ones. Spatial coordinates are defined in the topoSetDict file.

Thermal properties values of the air and of the solid walls are reported in

different dictionaries which must be created each time for different cases. As

for the air dictionary, it contains the file for defining the value of the

gravitation acceleration, which also in this case is modified in order to set the

desired value for the Rayleigh number, the turbulenceProperties file and the

RASProperties file. As for the turbulence models used, in this case only

standard 𝑘 − 휀 model with LR wall functions and Menter 𝑆𝑆𝑇 𝑘 − 𝜔 model

without wall functions are considered, from the consideration reported in

section A.3.3. Thermal properties are reported in the

thermophysicalProperties file present in each dictionary, where it is

necessary to define the additional equation of state used to close the

mathematical system, transport properties and thermodynamics properties.


141

Table A.18: Thermo-Physical Properties (300 𝐾)

Air AISI-316 Aluminium Copper Eq. of state perfectGas (at 1 𝑎𝑡𝑚) rhoConst rhoConst rhoConst

𝒄 (𝑱 𝒌𝒈−𝟏𝑲−𝟏) 1007 468 903 385

𝝆 (𝒌𝒈 𝒎−𝟑) 1.1614 8238 2702 8933

𝝀 (𝑾 𝒎−𝟏𝑲−𝟏) - 13.4 237 401

𝝁 (𝑷𝒂 𝒔) 1.846 × 10−5 - - -

Table A.18 summarizes the relevant information for the air as fluid and for

three different materials considered in this study for the solid region, namely:

AISI-316, pure Aluminium and pure Copper.

The IC and BC for the cavity is almost the same as the previous cases, the

only difference being the BC at the left and right sides. This time, the fixed

hot and cold temperatures are imposed at the left side of the left solid region

and at the right side of the right solid region, respectively, and at the

interface solid-fluid the turbulentTemperatureCoupledBaffleMixed BC is

applied. Solid regions are initially at a uniform temperature equal to that

imposed as external BC.

As far as the numerical setting is concerned, even in this case a dictionary

must be created for each region. As for the air, the tolerance is set to 10−7 for

all the physical variables, and the same relaxation factors used earlier are

imposed. The upwind discretization is employed as scheme for the divergence

terms in both laminar and turbulent flow regimes, while the default

discretization schemes are used for the other terms. For the solid regions,

default settings are used. At last, the maximum Courant number is set at 5 in

order to obtain sufficiently accurate solution and to reduce the computational

time.

A.4.2 Results

In this section, all the results obtained with the conjugate heat transfer solver

are summarized. In particular, the quantities reported are the same as those

considered in the previous sections. Simulations are conducted for 3000 𝑠𝑒𝑐

and 600 𝑠𝑒𝑐 for laminar and turbulent flow regime, respectively.

In Table A.19 is reported the grid-independent study for the case of laminar

flow regime (𝑅𝑎 = 106), with the RE between two successive grids, for all the

materials considered for the solid regions, while in Tables A.20 and A.21 the

same results are reported for the case at 𝑅𝑎 = 1.58 × 109 for 𝑘 − 휀 model and

𝑆𝑆𝑇 𝑘 − 𝜔 model, respectively. A comparison with the results obtained

without TI is shown in Tables A.22 and A.23, with RE.



numbers

142

Figures of temperature and velocity distributions are not reported since they

are very similar to those reported in previous sections.

Table A.19: Grid sensitivity for the TI case (𝑅𝑎 = 106)

AISI-316 Aluminium Copper

(𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎) (𝟏𝟎𝟎 × 𝟏𝟎𝟎) (𝟏𝟐𝟎 × 𝟏𝟐𝟎)

𝑼 68.22466 67.61403 67.61671 67.31519 67.97674 67.29375

RE / 0.895 % / 0.446 % / 1.005 %

𝒚 0.86 0.86 0.86 0.86 0.86 0.86

RE / 0 / 0 / 0

𝑽 219.1334 218.6188 218.7769 218.5674 219.2996 218.6997

RE / 0.235 % / 0.096 % / 0.274 %

𝒙 0.04 0.04 0.04 0.04 0.04 0.04

RE / 0 / 0 / 0

𝑵𝒖𝒎𝒂𝒙 17.84 17.32 17.56 17.51 17.61 17.62

RE / 2.915 % / 0.285 % / 0.057 %

𝒚𝒎𝒂𝒙 0.04 0.04 0.04 0.04 0.04 0.04

RE / 0 / 0 / 0

𝑵𝒖𝒎𝒊𝒏 0.9 0.84 0.85 0.84 0.85 0.84

RE / 6.667 % / 1.176 % / 1.176 %

𝒚𝒎𝒊𝒏 1 1 1 1 1 1

RE / 0 / 0 / 0

Table A.20: Grid sensitivity for the TI case (𝑅𝑎 = 1.58 × 109, 𝑘 − 휀 model)


(𝟏𝟔𝟎 × 𝟏𝟔𝟎) (𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟏𝟔𝟎 × 𝟏𝟔𝟎) (𝟏𝟖𝟎 × 𝟏𝟖𝟎) (𝟏𝟔𝟎 × 𝟏𝟔𝟎) (𝟏𝟖𝟎 × 𝟏𝟖𝟎)

𝑽 0.21032 0.21684 0.21259 0.21930 0.21291 0.21967

RE / 3.096 % / 3.157 % / 3.177 %

𝒙 0.007 0.006 0.007 0.006 0.007 0.006

RE / 14.286 % / 14.286 % / 14.286 %

𝒚+ (h.w.) 1.17457 1.02739 1.18018 1.03326 1.18121 1.00287

𝒚+ (v.w.) 4.72266 4.46285 4.74546 4.486 4.74864 4.48959

𝑵𝒖𝟏 𝟐⁄ 52.5882 54.6581 53.7484 55.7887 53.91 55.9712

RE / 3.739 % / 3.796 % / 3.823 %

Table A.21: TI results (𝑅𝑎 = 1.58 × 109, 𝑆𝑆𝑇 𝑘 − 𝜔 model)


𝑽 0.21032 0.21259 0.21291

𝒙 0.007 0.007 0.007

𝑵𝒖𝟏 𝟐⁄ 52.5882 53.7484 53.91


143

Table A.22: Comparison with previous data (𝑅𝑎 = 106)

AISI-316 Aluminium Copper No TI

𝑼 67.61403 67.31519 67.29375 64.87263

RE 4.226 % 3.765 % 3.732 % /

𝒚 0.86 0.86 0.86 0.85

RE 1.176 % 1.176 % 1.176 % /

𝑽 218.619 218.567 218.700 219.571

RE 0.433 % 0.457 % 0.397 % /

𝒙 0.04 0.04 0.04 0.04

RE 0 0 0 /

𝑵𝒖𝒎𝒂𝒙 17.32 17.51 17.62 17.6

RE 1.591 % 0.511 % 0.114 % /

𝒚𝒎𝒂𝒙 0.04 0.04 0.04 0.04

RE 0 0 0 /

𝑵𝒖𝒎𝒊𝒏 0.84 0.84 0.84 0.9

RE 6.667 % 6.667 % 6.667 % /

𝒚𝒎𝒊𝒏 1 1 1 0.99

RE 1.010 % 1.010 % 1.010 % /

Table A.23: Comparison with previous data (𝑅𝑎 = 1.58 × 109, 𝑘 − 휀 model)

AISI-316 Aluminium Copper No TI

𝒌 − 𝜺 𝑺𝑺𝑻 𝒌 −𝝎 𝒌 − 𝜺 𝑺𝑺𝑻 𝒌 −𝝎 𝒌 − 𝜺 𝑺𝑺𝑻 𝒌 −𝝎 𝒌 − 𝜺 𝑺𝑺𝑻 𝒌 − 𝝎

𝑽 0.2168 0.2438 0.2193 0.2456 0.2197 0.2461 0.2184 0.2421

RE 0.733 % 0.702 % 0.412 % 1.445 % 0.595 % 1.652 % / /

𝒙 0.006 0.007 0.006 0.007 0.006 0.007 0.006 0.007

RE 0 0 0 0 0 0 / / 𝑵𝒖𝟏 𝟐⁄ 54.658 56.771 55.789 57.466 55.971 57.625 58.031 59.470

RE 5.812 % 4.538 % 3.863 % 3.370 % 3.550 % 3.102 % / /

Following the same criteria used in the previous cases for the definition of

grid-independent solutions, from Table A.19 it can be noticed that RE of

velocity components and their locations are less than 1 % and the greatest

errors are the location of the minimum value of the Nusselt number for the

laminar case, coherently with what observed in the case without TI, while in

Table A.20 RE have almost the same value as those seen previously (see

Table A.11). About simulations with 𝑆𝑆𝑇 𝑘 − 𝜔 model, only one grid was

considered for computational cost reason.

As expected, for the examined system, the results do not differ from those of

the previous sections, at least within the level of detail of the adopted model,

as shown in Tables A.22 and A.23, with RE below 5 %. The greatest

differences are the x-component of the velocity, in the laminar case, and the

Nusselt number at the mid-height, in the turbulent case. The values of 𝑦+ are

almost equal to those obtained previously. This fact, in addition to the low RE

reported in Table A.23, implies that the presence of TI does not affect the

results also in case of turbulent flow regime.



numbers

144

As for the different wall materials considered, AISI-316 and Copper present

similar values for 𝜌𝑐 but different thermal conductivity (see Table A.18),

while Aluminium has an intermediate value for the thermal conductivity but

a lower 𝜌𝑐. In spite of these differences, the results obtained in the three

materials are very similar to each other.

A.5 Final remarks

In this Appendix, the well-known case of the buoyancy-driven cavity is

considered in order to familiarize with the OpenFOAM® code. The obtained

results are assessed on the basis of both numerical and experimental

benchmarks. Different flow regimes for natural convection are studied

considering a wide range of Rayleigh numbers. As shown by the temperature

and velocity profiles, the air moves along a more and more narrow boundary

layer where velocity and temperature change sharply, whereas in the core

region of the cavity the air is overall at rest. This is due to the increase of the

Rayleigh number, and then of the Grashof number, resulting in a flow regime

gradually more and more close to the turbulent region, and consequently it is

necessary to obtain very accurate solutions in the near-wall region. As for

laminar and transition flow regimes, a good agreement is obtained between

simulations and numerical benchmarks, even using simple numerical

schemes, thanks to the regularity of the motion in this range of Rayleigh

numbers. Less accurate results are obtained for turbulent flow regime.

Several problems must be faced when dealing with turbulent flows, especially

at low turbulence flow regime, namely: the lack of a universal turbulence

model that could accurately predict all flow variables; the near-wall

treatment, which represents one of the main sources of errors in numerical

simulations; difficulties and uncertainties which arise in experimental set-up.

Several RANS two-equations turbulence models and different near-wall

treatment are used in order to investigate how these choices influence the

results. In general, relative errors between simulations and the experimental

benchmark by Ampofo & Karayiannis (2003) are less than 10 %. In

particular, standard 𝑘 − 휀 with Low-Reynolds wall functions shows the best

results for the system considered in this study, predicting both velocity and

Nusselt number quite well. At last, thermal inertia are introduced in order to

show the chtSourceMultiRegionFoam conjugate heat transfer solver of the

OpenFOAM® code. For the considered system, no differences are expected

with respect to the case without thermal inertia, and this is confirmed a

References

145

posteriori by the obtained results, which do not differ from each other, at least

within the level of approximation of the adopted model.

As final remark, the OpenFOAM® code is reliable in simulating flows in

laminar and transition regimes, as demonstrated by the several numerical

benchmarks available in literature. Solutions for turbulent flow regimes can

be considered acceptable, even if many efforts are still necessary to improve

performances of numerical simulations, especially with more complex flows.

References


turbulent natural convection in an air filled square cavity. International




NJ, United States.

Corzo, S., Daminàn, S., Ramajo, D. & Nigro, N., 2011. Numerical Simulation

of Natural Circulation Phenomena. Asociasòn Argentina Mecànica

Computational, Volume XXX, pp. 277-296.

De Vahl Davis, G., 1983. Natural convection of air in a square cavity: a bench

mark numerical solution. International Journal for Numerical Methods in

Fluids, 3(3), pp. 249-264.



Tian, Y. S. & Karayiannis, T. G., 2000. Low turbulence natural convection in

an air filled square cavity, Part I: Thermal and fluid flow fields. Internation





Industries Inc.

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