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
iii
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
iv
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
v
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
vii
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.
ix
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.
xi
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
xiii
ESTRATTO IN ITALIANO
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
ESTRATTO IN ITALIANO
xiv
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.
ESTRATTO IN ITALIANO
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.
ESTRATTO IN ITALIANO
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⁄ ).
ESTRATTO IN ITALIANO
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)).
ESTRATTO IN ITALIANO
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.
ESTRATTO IN ITALIANO
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
ESTRATTO IN ITALIANO
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.
ESTRATTO IN ITALIANO
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.
ESTRATTO IN ITALIANO
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.
ESTRATTO IN ITALIANO
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 ℃.
ESTRATTO IN ITALIANO
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
ESTRATTO IN ITALIANO
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:
ESTRATTO IN ITALIANO
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.
ESTRATTO IN ITALIANO
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
una potenza di 10 𝑘𝑊.
(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.
ESTRATTO IN ITALIANO
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
una potenza di 10 𝑘𝑊.
ESTRATTO IN ITALIANO
xxix
(a)
(b)
Figura E.23: 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 2 𝑘𝑊.
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.
ESTRATTO IN ITALIANO
xxx
(a)
(b)
Figura E.24: 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 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.
ESTRATTO IN ITALIANO
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
FIGURA E.23: 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 2 𝑘𝑊. ................................... XXIX
FIGURA E.24: 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 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
FIGURE 2.8: COMPARISON OF THE VELOCITY DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 107: CASE (A)
𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; CASE (D)
𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ................................................................................................... 45
LIST OF FIGURES
xli
FIGURE 2.9: COMPARISON OF THE TEMPERATURE DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 107: CASE
(A) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; CASE
(D) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ..............................................................................................45
FIGURE 2.10: COMPARISON OF THE VELOCITY DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 108: CASE (A)
𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; CASE (D)
𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞. ....................................................................................................48
FIGURE 2.11: COMPARISON OF THE TEMPERATURE DISTRIBUTION FOR 𝑅𝑎𝑒𝑥𝑡 = 108: CASE
(A) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; CASE (B) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; CASE (C) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; CASE
(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
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. ........................... 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
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. ............................................................ 82
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 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. .................................... 82
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. .......................... 83
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 INERTIA AND USING THE 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE
MODEL. .................................................................................................................. 83
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 𝑆𝑆𝑇 𝑘 − 𝜔 TURBULENCE MODEL. .......................... 84
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 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
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. ......................................................... 86
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. ...................................... 86
LIST OF FIGURES
xliii
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. ....................................................................87
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 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
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. ..............................................92
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. ........................................................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
FIGURE 3.42: EXPERIMENTAL VS. SIMULATED TEMPERATURE (O-O AND CFD WITH
𝑆𝑆𝑇 𝑘 − 𝜔) TAKEN AT PROBE POSITION TC-5 FOR EXPERIMENT 1A. ........................93
FIGURE 3.43: EXPERIMENTAL VS. SIMULATED TEMPERATURE (O-O AND CFD WITH
𝑆𝑆𝑇 𝑘 − 𝜔) TAKEN AT PROBE POSITION TC-20 FOR EXPERIMENT 9. ........................93
FIGURE 3.44: EXPERIMENTAL VS. SIMULATED TEMPERATURE (O-O AND CFD WITH
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
FIGURE 4.10: STEADY-STATE RADIAL TEMPERATURE DISTRIBUTION TAKEN AT THE
COOLER SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF IHG AND 10 𝑘𝑊. . 110
FIGURE 4.11: STEADY-STATE RADIAL VELOCITY DISTRIBUTION TAKEN AT THE COOLER
SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF IHG AND 10 𝑘𝑊. .............. 110
FIGURE 4.12: MASS FLOW RATE TRANSIENT BEHAVIOUR OBTAINED WITH THE
OPENFOAM® CODE FOR THE IHG AND A-EHF CASES, AT 2 𝑘𝑊. ......................... 111
FIGURE 4.13: CFD VS. O-O MODELS FOR THE IHG (A) AND A-EHF (B) HEATING MODES,
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
OSCILLATION) AND 2 𝑘𝑊. ..................................................................................... 112
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. ..................................................................................... 113
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
OSCILLATION) AND 2 𝑘𝑊. ..................................................................................... 113
FIGURE 4.18: MASS FLOW RATE TRANSIENT BEHAVIOUR OBTAINED WITH THE
OPENFOAM® CODE FOR THE IHG AND A-EHF CASES, AT 0.5 𝑘𝑊. ....................... 113
FIGURE 4.19: CFD VS. O-O MODELS FOR THE IHG (A) AND A-EHF (B) HEATING MODES,
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
FIGURE 4.21: STEADY-STATE RADIAL TEMPERATURE DISTRIBUTION TAKEN AT THE
COOLER SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF A-EHF AND 0.5 𝑘𝑊.
............................................................................................................................. 115
FIGURE 4.22: STEADY-STATE RADIAL VELOCITY DISTRIBUTION TAKEN AT THE COOLER
SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF A-EHF AND 0.5 𝑘𝑊. .......... 115
FIGURE 4.23: STEADY-STATE RADIAL TEMPERATURE DISTRIBUTION TAKEN AT THE
COOLER SECTION (A) AND AT THE BOTTOM LEG (B) IN CASE OF IHG AND 0.5 𝑘𝑊. . 115
FIGURE 4.24: STEADY-STATE RADIAL VELOCITY DISTRIBUTION TAKEN AT THE COOLER
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.3: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 104. .................. 129
FIGURE A.4: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 105. .................. 129
FIGURE A.5: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 106. .................. 130
FIGURE A.6: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 107. .................. 132
FIGURE A.7: TEMPERATURE AND VELOCITY DISTRIBUTION FOR 𝑅𝑎 = 108. .................. 133
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.3: GRID SENSITIVITY 𝑅𝑎 = 104 ..................................................................... 129
TABLE A.4: GRID SENSITIVITY 𝑅𝑎 = 105 ..................................................................... 129
TABLE A.5: GRID SENSITIVITY 𝑅𝑎 = 106 ..................................................................... 130
TABLE A.6: COMPARISON WITH BENCHMARKS ............................................................ 131
TABLE A.7: GRID SENSITIVITY 𝑅𝑎 = 107 ..................................................................... 132
TABLE A.8: GRID SENSITIVITY 𝑅𝑎 = 108 ..................................................................... 133
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.
International Journal of Heat and Mass Transfer, 43(17), pp. 3027-3051.
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.
International Journal of Heat and Mass Transfer, 54(11), 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, 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 Turbulence: theory and models
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
CHAPTER 1: Brief overview of Computational Fluid Dynamics
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).
1.3 Turbulence: theory and models
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:
CHAPTER 1: Brief overview of Computational Fluid Dynamics
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).
1.3 Turbulence: theory and models
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)
CHAPTER 1: Brief overview of Computational Fluid Dynamics
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:
1.3 Turbulence: theory and models
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)
CHAPTER 1: Brief overview of Computational Fluid Dynamics
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)
1.3 Turbulence: theory and models
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)
CHAPTER 1: Brief overview of Computational Fluid Dynamics
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:
CHAPTER 1: Brief overview of Computational Fluid Dynamics
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
CHAPTER 1: Brief overview of Computational Fluid Dynamics
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.
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.
2.3 Modelling and implementation in OpenFOAM®
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.
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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 %
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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 %
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
38
Table 2.4: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100
(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 %
𝒚𝒎𝒊𝒏 0 0 0
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
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
40
Case (a) Case (b) Case (c) Case (d)
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⁄ = ∞.
Case (a) Case (b) Case (c) Case (d)
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
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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.
Table 2.7: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1
(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 %
𝒚𝒎𝒊𝒏 1 1 1
RE / 0 0
𝑵𝒖𝟏 𝟐⁄ 14.522 14.533 14.522
RE / 0.076 % 0.007 %
2.5 Results
43
Table 2.8: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100
(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 %
𝒚𝒎𝒊𝒏 0 0 0
RE / 0 0
𝑵𝒖𝟏 𝟐⁄ -21.246 -21.3263 -22.102
RE / 0.378 % 3.636 %
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
44
Table 2.9: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞
(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 %
Table 2.10: Comparison 𝑅𝑎 = 107
Case (a) Case (b) Case (c) Case (d)
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
Case (a) Case (b) Case (c) Case (d)
Figure 2.8: Comparison of the velocity distribution for 𝑅𝑎𝑒𝑥𝑡 = 10
7: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.
Case (a) Case (b) Case (c) Case (d)
Figure 2.9: Comparison of the temperature distribution for 𝑅𝑎𝑒𝑥𝑡 = 10
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).
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
46
Even in this case, no effects of IHG and of the Prandtl number on the flow
regime are found.
Table 2.11: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1
(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 %
𝒚𝒎𝒊𝒏 1 1 1
RE / 0 0
𝑵𝒖𝟏 𝟐⁄ 26.1093 26.1753 26.1907
RE / 0.253 % 0.059 %
Table 2.12: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100
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
Table 2.13: Grid Sensitivity 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞
(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 %
Table 2.14: Comparison 𝑅𝑎 = 108
Case (a) Case (b) Case (c) Case (d)
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
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
48
Case (a) Case (b) Case (c) Case (d)
Figure 2.10: Comparison of the velocity distribution for 𝑅𝑎𝑒𝑥𝑡 = 10
8: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.
Case (a) Case (b) Case (c) Case (d)
Figure 2.11: Comparison of the temperature distribution for 𝑅𝑎𝑒𝑥𝑡 = 10
8: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.
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).
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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
𝑻𝒊𝒏 (𝑲) 𝑻𝒉 (𝑲) 𝑻𝒄 (𝑲) 𝑳 (𝒎) 𝑾 (𝒎) 𝜷 (𝑲−𝟏) 𝝂 (𝒎𝟐𝒔−𝟏) 𝑷𝒓
300 320 280 0.75 1.5 3.4 × 10−3 1.59 × 10−5 0.71
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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)
Laminar Model (𝟏𝟒𝟎 × 𝟏𝟒𝟎)
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))
Laminar Model (𝟏𝟒𝟎 × 𝟏𝟒𝟎)
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))
Case (a) Case (b) Case (c) Case (d)
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
Case (a) Case (b) Case (c) Case (d)
Figure 2.15: Comparison of the velocity distribution for 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 10
9: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
54
Case (a) Case (b) Case (c) Case (d)
Figure 2.16: Comparison of the temperature distribution for 𝑅𝑎𝑒𝑥𝑡 = 1.58 × 10
9: case (a) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 0; case (b) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 1; case (c) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = 100; case (d) 𝑅𝑎𝑖𝑛𝑡 𝑅𝑎𝑒𝑥𝑡⁄ = ∞.
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
2.6 Effects of the Prandtl number on the flow regime
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.
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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.
2.6 Effects of the Prandtl number on the flow regime
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
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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
2.7 Effects of thermal inertia
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 - -
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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 %
2.7 Effects of thermal inertia
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.
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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
CHAPTER 2: Natural circulation of fluids characterized by an internal energy source
in a square cavity at different Rayleigh numbers
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
Ampofo, F. & Karayiannis, T. G., 2003. Experimental benchmark data for
turbulent natural convection in an air filled square cavity. Internation
Journal of Heat and Mass Transfer, 46(19), pp. 3551-3572.
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
Le Quéré, P., 1991. Accurate solutions to the square thermally heated driven
cavity at high Rayleigh. Computers & Fluids, 20(1), pp. 29-41.
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.
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.
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).
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
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;
3.3 Stability maps for single-phase natural circulation loops
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
𝜌𝑤𝑐𝑤𝜕𝑇𝑤,𝑜𝜕𝑡
=𝑇𝑤,𝑖 − 𝑇𝑤,𝑜
��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)
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
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.
3.4 Modelling and implementation in OpenFOAM®
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.
3.4 Modelling and implementation in OpenFOAM®
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.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
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
laminar and transition flow regimes and RASModel is set for turbulent flow
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.
3.4 Modelling and implementation in OpenFOAM®
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.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
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.
3.4 Modelling and implementation in OpenFOAM®
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
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
78
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
3.5 Numerical simulation results vs. experimental data
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.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
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.
3.5 Numerical simulation results vs. experimental data
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.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
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
𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.
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.
3.5 Numerical simulation results vs. experimental data
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
inertia and using the 𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.
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.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
loops
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
𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.
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
inertia and using the 𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.
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.
3.5 Numerical simulation results vs. experimental data
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
𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.
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 ℃.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
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.
3.5 Numerical simulation results vs. experimental data
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
𝑆𝑆𝑇 𝑘 − 𝜔 turbulence model.
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
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
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
3.5 Numerical simulation results vs. experimental data
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.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
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.
3.5 Numerical simulation results vs. experimental data
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).
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
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.
3.5 Numerical simulation results vs. experimental data
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.
Figure 3.44: Experimental vs. simulated
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.
Figure 3.46: Experimental vs. simulated
(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
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
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.
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.
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, 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.
OpenFOAM, 2016, http://www.openfoam.com/; http://www.openfoam.org/.
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.
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, Issue 145, pp.
108-125.
Ruiz, D., Cammi, A. & Luzzi, L., 2015. Dynamic stability of natural
circulation loops for single phase fluids with internal heat generation.
Chemical Engineering Science, Issue 126, pp. 573-583.
CHAPTER 3: Dynamic stability for single-phase natural circulation in rectangular
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.
Versteeg, H. & Malalasekera, W., 2007. An Introduction to Computational
Fluid Dynamics: The Finite Volume Method. s.l.:Pearson Education.
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.
Wilcox, D., 1993b. Turbulence Modelling for CFD. La Canada, CS: DCW
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.
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
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.
4.3 Stability maps for single-phase natural circulation loops
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)):
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
104
IHG A-EHF
𝜌𝑓∗𝑐𝑓
𝜕𝑇𝑓
𝜕𝑡+ 𝐺𝑐𝑓
𝜕𝑇𝑓
𝜕𝑠=
= −ℎ(𝑇𝑓 − 𝑇𝑤,𝑖)��𝑓
��𝑓+ 𝑞′′′
𝜌𝑓∗𝑐𝑓
𝜕𝑇𝑓
𝜕𝑡+ 𝐺𝑐𝑓
𝜕𝑇𝑓
𝜕𝑠= −ℎ(𝑇𝑓 − 𝑇𝑤,𝑖)
��𝑓
��𝑓 (1)
{
𝑇𝑤,𝑜 = 𝑇𝑐 (𝑎)
𝜌𝑤𝑐𝑤𝜕𝑇𝑤,𝑜𝜕𝑡
=𝑇𝑤,𝑖 − 𝑇𝑤,𝑜
��𝑤,𝑜��𝑤+��𝑤,𝑜
��𝑤,𝑜𝑞′′ (𝑏)
𝜌𝑤𝑐𝑤𝜕𝑇𝑤,𝑜𝜕𝑡
=𝑇𝑤,𝑖 − 𝑇𝑤,𝑜
��𝑤,𝑜��𝑤 (𝑐)
{
𝑇𝑤,𝑜 = 𝑇𝑐 (𝑎)
𝜌𝑤𝑐𝑤𝜕𝑇𝑤,𝑜𝜕𝑡
=𝑇𝑤,𝑖 − 𝑇𝑤,𝑜
��𝑤,𝑜��𝑤+��𝑤,𝑜
��𝑤,𝑜𝑞′′ (𝑏)
𝜌𝑤𝑐𝑤𝜕𝑇𝑤,𝑜𝜕𝑡
=𝑇𝑤,𝑖 − 𝑇𝑤,𝑜
��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.
4.4 Modelling and implementation in OpenFOAM®
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).
4.4 Modelling and implementation in OpenFOAM®
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 %
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
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.
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
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.
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
110
(a) (b)
Figure 4.10: Steady-state radial temperature distribution taken at the cooler section
(a) and at the bottom leg (b) in case of IHG and 10 𝑘𝑊.
(a) (b)
Figure 4.11: Steady-state radial velocity distribution taken at the cooler section (a)
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)
Figure 4.13: CFD vs. O-O models for the IHG (a) and A-EHF (b) heating modes, at
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.
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
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.
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
114
(a)
(b)
Figure 4.19: CFD vs. O-O models for the IHG (a) and A-EHF (b) heating modes, at
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)
Figure 4.21: Steady-state radial temperature distribution taken at the cooler section
(a) and at the bottom leg (b) in case of A-EHF and 0.5 𝑘𝑊.
(a) (b)
Figure 4.22: Steady-state radial velocity distribution taken at the cooler section (a)
and at the bottom leg (b) in case of A-EHF and 0.5 𝑘𝑊.
(a) (b)
Figure 4.23: Steady-state radial temperature distribution taken at the cooler section
(a) and at the bottom leg (b) in case of IHG and 0.5 𝑘𝑊.
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
116
(a) (b)
Figure 4.24: Steady-state radial velocity distribution taken at the cooler section (a)
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
CHAPTER 4: Dynamic stability for natural circulation with internally heated fluid:
the DYNASTY facility
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.
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).
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.
OpenFOAM, 2016, http://www.openfoam.com/; http://www.openfoam.org/.
Pini, A., Cammi, A. & Luzzi, L., 2016. Analytical and numerical investigation
of the heat exchange effect on the dynamic behaviour of natural circulation
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.
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.
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.
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
𝑻𝒊𝒏 (𝑲) 𝑻𝒉 (𝑲) 𝑻𝒄 (𝑲) 𝑳 (𝒎) 𝑾 (𝒎) 𝜷 (𝑲−𝟏) 𝝂 (𝒎𝟐𝒔−𝟏) 𝑷𝒓
300.5 301 300 1 0.1 3 × 10−3 10−5 0.71
Turbulent flow regimes
𝑻𝒊𝒏 (𝑲) 𝑻𝒉 (𝑲) 𝑻𝒄 (𝑲) 𝑳 (𝒎) 𝑾 (𝒎) 𝜷 (𝑲−𝟏) 𝝂 (𝒎𝟐𝒔−𝟏) 𝑷𝒓
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
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 10−7 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.
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
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
Table A.3: Grid Sensitivity 𝑅𝑎 = 104
(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
Figure A.3: Temperature and velocity distribution for 𝑅𝑎 = 104.
Table A.4: Grid Sensitivity 𝑅𝑎 = 105
(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
Figure A.4: Temperature and velocity distribution for 𝑅𝑎 = 105.
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
130
Table A.5: Grid Sensitivity 𝑅𝑎 = 106
(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
Figure A.5: Temperature and velocity distribution for 𝑅𝑎 = 106.
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
This work Corzo et al.
(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 %
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
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.
Table A.7: Grid Sensitivity 𝑅𝑎 = 107
(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 %
Figure A.6: Temperature and velocity distribution for 𝑅𝑎 = 107.
A.3 Results
133
Table A.8: Grid Sensitivity 𝑅𝑎 = 108
(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 %
Figure A.7: Temperature and velocity distribution for 𝑅𝑎 = 108.
Table A.9: Comparison with benchmarks
𝑹𝒂 = 𝟏𝟎𝟕 𝑹𝒂 = 𝟏𝟎𝟖
This work Corzo et al.
(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
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
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 %
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
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.
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
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.
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
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.
A.4 Effects of Thermal Inertia
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.
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
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)
AISI-316 Aluminium Copper
(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)
AISI-316 Aluminium Copper
𝑽 0.21032 0.21259 0.21291
𝒙 0.007 0.007 0.007
𝑵𝒖𝟏 𝟐⁄ 52.5882 53.7484 53.91
A.4 Effects of Thermal Inertia
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.
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
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
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.
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.
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.
Le Quéré, P., 1991. Accurate solutions to the square thermally heated driven
cavity at high Rayleigh. Computers & Fluids, 20(1), pp. 29-41.
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
Journal of Heat and Mass Transfer, 43(6), pp. 849-866.
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.