Direct and large Eddy simulation of turbulent flow in a cylindrical pipe geometry

TECH1JSCHE UNIVERSITEIT Laboratorium voor Schoepshydromechanl

Archief

Mekeiweg 2,2628 CD Daift

TcL O15.7ßI3. Feic O15.7I1S

Direct and Large Eddy Simulátion

of Tùrbulent Flow in a

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Direct and Large. Eddy Simulation

f Turbulent Flow in

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Cylindrical Pipe Geometry

PROEFSCHRIFT

Ter 'verkrijging van dè graad van doctor aan de Tec]flithe Universiteit Deift, op gezag van de

Rector Magnificus, prof. ir. K.F Wakker, in

het openbaar te verdedigén voór een commissie aangewezen door het College van Dekanen op

donderdag 10 februari 1994 orn 16.00 uur

door

Jacobus Gerardus Mária Eggels

bouwkundig ingeniéur,

geboren te Breda.

Dit proefschrift is goedgekeurd door de promotoren: Prof. dr. ir. F.T.M. Nieuwstadt

Prof. dr. ir. P. Wesseling

CIP-DATA KONINKLLJKE BIBLIOTHEEK, DEN HAAG Eggels, J.G.M.

Direct and Large Eddy Simulation of Turbulent Flow in a Cylindrical Pipe Geometiy J.G.M. Eggels - Dèlft;: Deift University Press. - Ill.

Thesis Delft University of Technology,

1994. With Summary. - With ref.

ISBN 90-6275-940-8 NUGI 834

Subject headings computational fluid dynamics / numerical simulation / turbulent flow

Published and distributed by: Delft University Press Stevinweg i 2628 CN Delft The Netherlands Phone (31)-15-783254 Fax (31)-15-781661 Copyright ©1994 by J.G.M. Eggels All rights reserved.

No part of the material protected by this copyright notice may be reproduced or utilized

inany form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the publisher. Printed in the Netherlands.

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Contents

Summary

iii ZusamxneÁifassung

y

Saïneñváttiiig

1 Introduction

i

1.1 Whr Turbulence? .

:.

i

1.2 Numerical Simulations 2

1.3 Outline of this thesis . .

.

4

2 Numerical Simulation of Turbulence

9

2.1 Introduction to Turbulence Theory 9

2.2 Principles of Numerical Simulations 12

2.2.1 Introduction

.

. . 12

2.2.2 Direct Numerical Simulation 13

2.2.3 Large-Eddy Simulation 5

2.2.4 Reynolds-averaged Modelling

...

18

3 Computational Techniques

21

3.1 IntrOduction 21

3.2 Turbulence Modelling . - 22

3.2.1 Turbulence Modelling in LES 22

3.2.1.1 Details of SGS Pararneterization 22

3.2.1.2 Discussion on SGS Parameteri.zation 28

3.2.2 'Dïrbulence Modelling in RaM . . . 30

3.3 Numerical Techniques 35

3.3.1

Spatial Discretization ...

36

3.3.2 Temporal Discretization and Numerical Solution Procedure . 40

3.3.3 Boundary Conditions and Forcing 43 3.3.4 Initial Conditions and Data Processing 46

u

4 Numerical Simulation of Turbulent Flow in Non-rotating Pipes

51

4.1 Introduction 52

4.2 Direct Numerical Simulation 53

4.2.1 Description of DNS Computation 53

4.2.2 Mean Flow Quantities 5? 4.2.3 Turbulence Statistics 60

4.2.4 Energy Budgets of the Reynolds Stress Components 65

4.2.5 Conclusions on DNS ₆₉ 4.2.6 Further Investigations Using the DNS Database . . 69

4.3 Large Eddy Simulations 70

4.3.1 Description of LES Computations 7()

4.3.2 Mean Flow Quantities 77 4.3.3 Türbulence Statistics 81

4.3.4 Energy Budgets of the Reynolds Stress Components 89 4.3.5 Conclusions on LES 91

4.4 Evaluation of Pressure-Strain Approximations 95

5 Numerical Sinulatiòn of Thrbilent Flow in a Rotating Pipe

103

5.1 Introduction . 104

5.2 Descriptioi of Simulation 105

5.3 Results 107

5.3.1 Mean Flow Quantities 107

5.3.2 Tuibulence Statistics 109

5.3.3 Energy Budgets of the Reynolds Stress Components 114

5.3.4 Conclusions .. 119

5.4 Evaluation of Pressure-Strain Approximations

...120

6 Conclusions 125

A Frequency AnalyÈis of the Fhite Differénce Schemes

131

B Analysis of the Asselin Filter

137

C Energy Budgets of the Reynolds Stress Components

139

List of Symbols

142

Acknowledgements ₁₄₈

Summary iii

Summary

Turbulent flows are not only of great practical importance but also of considerable funda-mental interest, because turbulence is one of the phenomena in nature which are still not completely understood. In this thesis, applications of numerical simulations are repotted as a tool to investigate turbulent flows in a cylindrical pipe geometry.

In Direct Numerical Simi.ilation (DNS), the governing Navier-Stokes equations are directly sòlved on a fine spatial and temporal grid in order to resolve ail scales of turbulent

motion. No assumptions arê included to model turbulent fluctuations. Because of the rapidly growing extent of scales with increasing Reynolds number, DNS is restricted

to low-Reynolds number flows in fairly simple geometries. In Large-Eddy Simulation (LES), the Navier-Stokes equations are spatially filtered to separate small-scale turbulent motions (the called SubGrid-Scale (SGS) stresses) from large-scale. motions (the so-called resolved or Grid-Scale (GS) stresses). The GS motions are resolved explicitly in

space and time whereas the influence of removed SGS motions on GS fluctuations is

modelled (SGS parameterization). In most of the LES computations reported in this

thesis, the Smagorinsky model is used for this purpose. In earlier studies, this model has been found to perform satisfactory when used with an additional wall damping function to guarantee proper behaviour of the SGS stresses towards the wall. A minor modification of the Smagorinsky model is suggested in this thesis which provides the correct behaviour of the SGS stresses towards the wall without using a wall damping function. The 'modified' SGS model performs reasonably well, but not yet completely satisfactory in the near-wall

region.

The governing equations (in a filtered form in case of LES) are formulated in a cylin-drical coordinate system and spatially discretized using the finite volume approach. This approach has the advantage that the singularity of the equations at the centerline of the system automatically vanishes. The major consequence of the application of the

cylin-drical coordinate system is the small gridspacing in circumferential direction near the

centerline. It imposes a severe limitation on the allowable time-step in case explicit time integration schemes are utilized. This problem is overcome using a combination of im-plicit and exim-plicit time integration schemes which eventually allows a time-step at least one order of magnitude larger compared to a fully explicit approach, while keeping the computational cost about as low as for a fully explicit method.

First, we have applied DNS and LES to fully developed turbulent flow in a straight pipe at low and moderate Reynolds numbers, i.e. Re = 7000 (DNS) and 50000 (LES),

based on centerline velocity and pipe diameter. The purpose of these simulations is twofold: a) validation of the performance of the employed numerical techniques (via

DNS) and the SGS parameterization (via LES) by means of simulations of standard well-known pipe flows, and b) generation of detailed databases to support fundamental and

practical turbulence research. The DNS results show an excellent agreement with the

experimental data, which confirms that the employed numerical techniques are adequate

for such simulations of turbulent flows. The overall agreement between LES and the

iv Summary

It is demonstrated that these deviations should be attributed to deficiencies of the SGS parameterization near the wall (e.g. to the specific shape of the wall damping fUnction). For LES of wall-bounded shear-driven turbulent flows as reported in this thesis, sufficient spatial resolution appears to be 'a must'. In terms of the ratio i/is with i a typical length scale of the GS motions and i a characteristic length scale of the largest SGS motions, it

is shown that one needs at least i/is > 2 in order to obtain realistic flow statistics from

LES (a simulation with i/is 1.1 failed to produce correct results).

Since the DNS and LES databases contain 3-dimensional time-dependent velocity and

pressure fields, these data are very useful for all kind of turbulence research studies. Several of such studies are reported briefly in this thesis. In one of these, the DNS

and LES data are used to compute the radial distributions of the coefficients involved in pressure-strain closure approximations of conventional Reynolds stress turbulence models.

It is found that the coefficient c1 in the Rotta hypothesis varies with r and cannot be

considered constant throughout the cross-section. On the other hand, the coefficients in

the 'rapid' part of the pressure-strain model appear to be reasonably constant in radial direction. Théir values, however, depend on Reynolds number, e.g. the coefficient C'

equals approximately 0.56 at the low-Reynolds number vs. 0.33 at the moderate. Next, we have considered the turbulent flow in an axially rotating pipe by means of LES. This flow configuration is one of the most simple rotating flows in which the phe-nomena peculiar for rotation are already present. Pipe wall rotation acts as a stabilizing body force which suppresses turbulence fluctuations in radial and axial directions. As a result, the turbulent shear stress is reduced which induces a deformation of the mean axial velocity profile. The ratio of bulk velocity and wall shear stress velocity increases which is characterized by a reduction of the skin frition coefficient. The mean circumferential velocity profile is close to parabolic, except jn the vicinity of the wall. The results ob-tained from LES confirm the experimental observations and provide a good quantitative agreement with the measurements. Due to the application of the gradient hypothesis, the SGS shear stresses .re always oriented in the direction of gradient transport, but in the rotating flow, counter-gradient transport is also observed. TherefOre, the application of the gradient hypothesis as part of the SGS parameterization is not optimal to model the SGS shear stresses in the rotating, pipe flow.

Similarly as for the non-rotating pipe flow, the LES data for the rotating flow are used to compute the coefficients in the pressure-strain closure approximation. The coefficient

C1 in the Rotta hyothesis now becomes negative throughout the cross-section. There is some indication hat this result is related to the fact that the Rotta hypothesis is not the dominating tern in the closure approximation. This role must be attributed to the 'rapid' part of the bressure-strain model. The coefficients of the 'rapid' part are again

reasonably constant in radial direction, but their values deviate from those obtained for the non-rotating flow. Hence, it appears that the values of the coefficients in Reynolds stress turbulence models cannot be considered universal, since they depend on the flow problem -and Reynlds number- considered.

Zusammenfassung y

Zusammenfassung1

Turbulente Strömungen sind nicht nur von großer praktischer Bedeutung sondern auch von

besonderem wissenschaftlichen Interesse, da die Turbulenz einer der noch nicht vollständig

verstandenen Phänomene in der Natur darstellt. Die vorliegende Arbeit stützt sich auf

die numerische Simulation, um mit diesem Hilfsmittel die turbulente Strömung durch ein zylindrisches Rohr zu untersuchen.

In der Direkten Numerischen Simulation (DNS) werden die Navier-Stokes Gleichungen

direkt gelöst. Das hierzu verwendete Rechengitter ist mit seiner feinen räumlichen und zeitlichen Auflösung in dér Lage, alle Längenmaße der Turbulenz zu erfassen. Somit sind keine' Annahmen zur Modellierung der turbulenten Schwankungen nötig. Aufgrund des starken Anstiegs des Skalenbereichs mit zunehmender Reynoldszahl und unter Berück-sichtigung begrenzter Computerkapazitäten, ist die DNS auf Strömungen mit niederen Reynoldszahlen und relativ einfachen Geometrien beschränkt. In der

Grobstruktursimula-tion (LES) werden die Navier-Stokes Gleichungen räumlich gefiltert und damit kleinskalige

Turbulenzbewegungen (die sogenannten SGS-Spannungen) von großskaligen Vorgängen (den sogenannten GS-Spannungen) getrennt. Die letzteren lassen sich explizit räumlich und zeitlich auflösen, wohingegen der Einfluß der Feinstruktur auf die Grobstruktur mod-elliert wird (Feinstrukturmodellierung). In den meisten Grobstruktursimulationen der vorliegenden Arbeit wurde hierzu das Smagorinsky-Modell verwendet. In früheren Unter-suchungen hat dieses Modell seine Leistungsfähigkeit bewiésen, obwohl eine zusätzliche Dämpfungsfunktion benötigt wird, um ein richtiges Verhalten der SGS-Spannungen in Wandnähe zu gewährleisten. In der vorliegenden Arbeit wird eine geringfügige Anderung des Smagorinsky-Modells vorgeschlagen, das damit die Feinstrukturspannungen in Wand-nähe ohne zusätzliche Dämpfungsfunktion richtig beschreibt. Das so modifizierte Modell arbeitet gut, ergibt aber im wandnahen Beréich immer noch keine endgültig zufrieden-stellenden Ergebnisse.

Die Ausgangsgleichungen (für die LES werden die Gleichungen in der gefilterten Form verwendet), formuliert für ein Zylinderkoordinatensystem, werden räumlich gemäß einem Fiñite-Volumen Ansatz diskretisiert. Dieser Ansatz hat den Vorteil, daß die

Singu-larität des Koordinatensystems auf der Rohrachse automatisch verschwindet. Die Ver-wendung eines Zylinderkoordinatensystems führt andererseits im Bereich der Rohrachse zu sehr kleinen Gitterweiten in Umfangsrichtung. Bei der Verwendung eines expliziten Zeitintegrationsverfahrens bedeutet dies eine harte Restriktion des maximal zulässigen Zeitschntts Emen Ausweg aus diesem Problem bietet eine Kombination aus implizitem und explizitem Zeitintegratiònsverfahren, die im Vergleich zum vollexpliziten Verfahren 'einen mindestens eine Größenordnung größeren Zeitschritt ermöglicht.

Im ersten Schritt werden mit Hilfe von DNS und LES vollentwickelte turbUlente Rohrströmungen bei niederer bzw. hoher Reynoldszahl durchgeführt (Re=7000 bzw.

50000, gebildet mit der Geschwindigkeit auf der Rohrachse und dem Rohrdurchmesser). Mit diesen Simulationen werden zwei Ziele verfolgt: a) Da mit der Rohrströmung eine

'Friédemann Unger of the Technical University of Munich (Germany) is gratefully acknowledged for translating this Zusammenfassung from the original English Summary.

vi Zusammenfassung

der am besten experimentell und theoretisch untersuchten Strömungen vorliegt, kann mit Hilfe der DNS die Leistungsfähigkeit der angewandten numerischen Methode und mit-tels der, LES die Feinstrukturmodellierung überprüft werden. b) Mit den Simulationen können andererseits detaillierte Datenbasen zur Unterstützung theoretischer und praktis-cher Turbulenzforschung bereitgestellt werden. Die hervorragende Ubereinstimmung der DNS-Ergebnisse mit dem Experiment bestätigt, daß die angewandte numerische Methode gut zur Simulation cieser Art turbulenter Strömungen geeignet ist. Insgesamt ist auch die Ubereinstimmung dr LES-Ergebnisse mit dem Experiment gut, obwohl sich im wandna-hen Bereich Abweichungen beobachten lassen. Es wird gezeigt, daß diese Abweichungen aufgrund der unzulänglichen Feinstrukturmodellierung im wandnahen Bereich entstehen (z.B., der spezielle Verlauf der Dämpfungsfunktion). Für die LES wanddominierter turbu-lenter Stherströmunen, wie sie in der vorliegenden Arbeit durchgeführt wurden, scheint eme ausreichende räimliche Auflösung em 'Muß' zu sein. Bezogen auf das Verhältnis i/ij,

mit I als typischem Liängenmaß der Grobstruktur und i als charakteristischem Längenmaß der größten Feinstrikturbewegungen, wird demonstriert, daß realistische Grobstruktur-simulationen nur für den Fall l/i > 2 durchgeführt werden können (eine Simulation mit

l/i

1.1 führte zu schlechten statistischen Ergebnissen).

Die mit Hilfe der DNS und LES erstellten Datenbasen enthalten die dreidimension-alen und zeitabhänigen Geschwindigkeits- und Druckfelder und sind damit von großem Interesse für Turbulnzforschungen. Einige dieser möglichen Untersuchungen sind in der vorliegenden Arbeit ufgeführt. SO wurden z.B. die DNS- und LES-Daten zur Berechnung des radialen Verlaufs der Koeffizienten benützt, die im Druck-Sther-Schließungsansatz von Reynoldsspannungs-Modellen verwendet werden. Es konnte nachgewiesen werden, daß der Koeffizient Çi der Rotta-Hypothese über den Radius r variiert und damit nicht als Konstante aufg4aßt werden kann. Andererseits sind die Koeffizienten im 'rapid' Teil des Schließungsansatzes nahezu konstant über den Radius, zeigen aber eine Abhängigkeit von der ReynoldszalI. (z.B. ergibt sich C' zu 0.56 für niedere bzw. 0.33 für höhere

Reynoldszahlen).

Weiterhin wurden mit Hilfe von Grobstruktursimulationen turbulente Strömungen in einem um die Rohrachse rotierenden Rohr durchgeführt. Diese Anordnung ist eine der einfachsten rotierenden Strömungen, in der die besonderen Phänomene dieser Strömungen untersucht werden lnnen. Die Rotation der Rohrwand wirkt ähnlich einer stabilisieren-den Körperkraft, die turbulente Fluktuationen in radialer und axialer Richtung unterdrückt. Als Folge nimmt die Reynoldsspannung ab, was letztendlich zu einer Deformation des mittleren Geschwindigkeitsproflls führt. Das Verhältnis zwischen der mittleren

Durch-flußgeschwindigkeit ùnd der Wandschubspannungsgeschwindigkéit stéigt an, was sich auch

in einer Abnahme des Reibungskoeffizienten äußert. Mit Ausnahme der unmittelbaren Umgebung der Rohi1achse ist das mittlere Geschwindigkeitsprofil in Umfangsrichtung na-hezu parabolisch. Die Ergebnisse der LES bestätigen die experimentellen Beobachtun-gen und stimmen qiantitativ gut mit den MessunBeobachtun-gen überein. Es konnte nachgewiesen werden, daß die im Smagorinsky-Modell verwendete Gradientenhypothese nicht zur Para-metrisierung der Feinstrukturspannungen herangezogen werden kann. Als Folge dieser Hypothese sind die Feinstrukturspannungen immer in Richtung des Gradiententransports

Zusammenfassung vii

ausgeñchtet, wohingegen in rotierenden Strömungen aucbFeinstrukturspannungenin ent-gegengesetzter Richtung beobachtet werden.

Entsprechend den Untersuchungen. nicht-rotierender Strömungen, wurden die. LES-Daten zur Berechnung des Koeffizienten im Druçk-Scher-Schließungsans3tz verwendet; Der Koeffizient C1 in der Rotta Hypothese nimmt negative Werte uber den gesamten

Radius än. Es gibt Grund zur Annahme, dieses Ergebnis der Tatsache zuzuschreiben,

daß die Rotta-Hypothese nicht den dominierenden Term im Schließungsansatz darstellt Diese Rolle kommt dem 'rapid Teil des Schließungsansatzes zu Die Konstanten des 'rapid' Teils sind wiederum nahezu konstant uber den Radius, erreichen aber Werte, die sich deutlich von der nicht-rotierenden Strömung unterscheiden

viii Samen vatting

Sainenvatting

Turbulente stromingen zijn niet alleen van groot praktisch belang maar bovendien vanuit fúndarnenteel oogpunt interessant, omdat turbulentie nog steeds niet volledig begrepen wordt. In dit proefschrift worden toepassingen van numerieke simulaties beschreven met als doel turbulènte stromingen in een cylindervormige geometrie (pup) te bestuderen.

In een Direkte Numerieke Sirnulatie (DNS), worden the Navier-Stokes vergelijkingen direkt opgelost op een fijnmazig netwerk in ruimte en tijd. Alle schalen van de turbu-lente fluctuatiés worden expliciet meegenomen in de sirnulatie met als gevoig dat geen enkele vorm van turbulentie modellering nodig is. De range van deze schalen groeit echter snel met toenemend Reynolds getal waardoor een enorme computer capaciteit nodig is voor sirnulaties van strorningen bij een hoog Reynolds getal. Door beperkingen aan de computer capaciteit is DNS alleen toepasbaar voor lage Reynolds getallen en voor stro-mingen in relatief eenvoudige geornetrieön. In een Large-Eddy Simulatie (LES), worden de Navier-Stokes vegelijkingen eerst ruirntelijk gefilterd orn de kleinschalige turbulente bewegingen te verwijderen. De gefilterde vergelijkingen beschrijven enkel de evolutie van de grootschalige bewegingen of 'wervelingen' (vandaar de naarn "Large-Eddy" Simulatie)

De invloed van de weggefilterde kleinschalige turbulente bewegingen op de grotere schalen moet nu in rekening worden gebracht met behulp van een zogenaarnd SubGrid-Scale (SGS)

model. Voor de meeste sirnulaties beschreven in dit proefschrift is het Smagorinsky model gebruikt als SGS model. Dit is een vrij eenvoudig model, maar toepassingen in eerdere

imulaties hebben aangetoond dat dit model desalniettemin naar tevredenheid function-eert. In de nabijheid van een wand is wel een wand-dempingsfunctie nodig orn het juiste gedrag van de kleinschalige bewegingen te verkrijgen. In dit proefschrift is een kleine wi-jziging van het standaard Smagorinsky model voorgesteld, waardoor het gebruik van de

wand-dempingsfunctie overbodig wordt. Dit gewijzigde model werkt redelijk goed, maar behoeft wel nog verbetering nabij de wand.

De differentiaalvergelijkingen (in een gefilterde vorm in het geval van LES) zijn gefor-muleerd iÌi cylinder coördinaten en ruimtelijk gediscretiseerd met behulp van de eindige

volume methode. Deze methode heeft de prettige eigenschap dat de singulariteit van

de vergelijkingen op de as van het coördinatensysteem automatisch verdwijnt. Als gevolg van het gebruik van cylinder coördinaten wordt de roosterafstand in omtreksrichting nabij de as erg klein. Dit heeft een belangrijke consequentie voor de te verwachten rekentijd van de simulatie. Omdat we veelal gebruik maken van zogenaamde expliciete discreti-satie schema's, is de toelaatbare tijdstap beperkt orn het rekenproces stabiel te houden. Aangezien deze tijdstap afhangt van de roosterafstand, wordt de toelaatbare tijdstap zeer klein als gevolg van de kleine roosterafstand nabij de as. Dit probleem is opgelost door gedeeltelijk gebruik te maken van impliciete discretisatie schema's die onvoorwaardelijk stabiel zijn. De toelaatbare tijdstap kan hierdoor, bij een fijne ruimtelijke resolutie, mm-stens een orde van grootte groter zijn dan bij gebruik van uitsluitend expliciete discretisatie

schema's.

Allereerst zijn DNS en LES berekeningen uitgevoerd voor volledig ontwikkelde turbu-lente stromingen door een rechte pijp. Het Reynolds getal in deze simulaties, gebaseerd

Sam envatting ix

op de gemiddelde sneiheid op de as van de pijp en de pijpdiameter, bedraagt 7000 voor de DNS en 50000 voor de LES. Het doe! van deze simulaties is tweeledig: a) bepaling van de geschiktheid van de numerieke technieken toegepast in de code (d.rn.v. DNS) en van de SGS modellering (d.rn.v. LES) door het uitvoeren van imuiaties van standaard pup-stromingen, en b) produktie van gedetailleerde datasets ter ondersteuning van algemeen turbulentie onderzoek. De DNS resultaten vertonen een uitstekende overeenkomst met experimentele data, hetgeen aantoont dat de toegepaste numerieke technieken geschikt

zijn voor dit soort simulaties. De overeenkomst tussen LES en experimenten is goed,

ofschoon er enkele afwijkingen zijn nabij de wand. Deze afwijkingen zijn het gevoig van een niet geheel juiste SGS modellering nabij de wand (b.v. de wand-dempingsfunctie geeft onvoldoende reductie van de SGS schuifspanning waardoor het gemiddelde snelheid-sprofiel wordt beïnvloed). Voor LES berekeningen van turbulente wandstromingen waarin

de turbulentie door afschuiving wordt gegenereerd (alleen dit type stromingen worden in dit proefschrift beschouwd), blijkt een fijne ruirntelijke resolutie van groot belang. In dit proefschrift wordt deze qualitatieve uitspraak gequantificeerd d.m.v. de parameter i/ij. Deze parameter geeft de verhouding weer tussen de karakteristieke lengteschaal i van de grootschalige turbulente bewegingen en die van de kleinschalige bewegingen, ij. De

ver-houding i/1, moet voldoende groot zijn (typisch, i/ij > 2) orn realistische resultaten te

verkrijgen uit een LES berekening (een sirnulatie met i/is 1.1, bleek niet in staat juiste resultaten te produceren).

Orndat de DNS en LES datasets 3-dimensionale, tijdafhankelijke, sneiheid- en drukvel-den bevatten, zijn deze data uitermate geschikt voor velerlei turbulentie onderzoek. Ver-schillende van zulke toepassingen zijn beschreven in dit proefschrift. In een hiervan zijn de DNS en LES data gebruikt orn de radiële verdeling van de coèfficiënten in de

'pressure-strain' sluitingsrelaties van Reynolds-spanning turbulentie modellen te bepalen. De coëfficiënt C1 in de Rotta hypothese blijkt niet constant voor alle r, maar af te nemen naar de wand toe. De coëfficiënten in de 'rapid' bijdrage van het 'pressure-strain' model,

blijken daarentegen redelijk constant in radiële richting, maar wel te variëren met bet

Reynolds getal. (b.v. voor de coëfficiënt C' werd de waarde 0.56 gevonden bij het lage Reynolds getal tegen 0.33 voor bet hogere Reynolds getal).

Na de simulaties van de standaard pijpstroming is een LES berekening uitgevoerd van de turbulente stroming in een axiaal roterende pijp. De strorning in deze configuratie kan worden gekarakteriseerd als een eenvoudige roterende stroming, waarin sommige effecten

als gevolg van rotatie reeds aanwezig zijn. De rotatie van de pijpwand heeft een

sta-biliserende invloed op de turbulente fluctuaties in radiële en axiale richtingen. Hierdoor wordt de turbulente schuifspanning verminderd, hetgeen aanleiding geeft tot een veran-dering van het gemiddelde axiale snelheidsprofiel. De verhouding van bulksnelheid (dit is de gemiddelde snelheid evenredig met de volumestroom door de pup) en wandschuif-snelheid neemt toe, waardoor de wrijvingscoëfficiënt afneemt. De gerniddelde sneiheid

in omtreksrichting blijkt nagenoeg parabolisch te zijn, behalve in de nabijheid van de

wand. De resultaten van de LES berekening komen goed overeen met de verschillende experimentele waarnemingen. De gradient hypothese, waarop het Smagorinsky model is gebaseerd, blijkt echter niet geheel meer te voldoen voor de roterende stroming. Deze

x Samen vattin,g

gradient hypothese schrijft voor dat de SGS schuifspanningen altijd georiënteerd zijn in de richting van gradient transport, terwiji in de roterende stroming ook transport tegen de gradient in optreedt.

Net als voor de standaard pijpstroming zijn de LES data van de roterende stroming ge-bruikt orn de coefficienten in de 'pressure-strain' sluitingsrelaties te bepalen De coefficient C1 in de Rotta. hypothese blijkt negatief te zijn voor de gehele doorsnede van de pijp. Er

bestaat de indruk dat dit resultaat te maken heeft met het feit dat de Rotta hypothese

niet de belangrijkste bijdrage in de sluitingsrelatie is. Deze rol moet worden toebedeeld

aan de 'rapid' bijdrage in het 'pressure-strain' model. De coéfficiënten in deze 'rapid'

bijdrage blijken nog steeds rêdelijk constant in radiële richting, maar wel aanzienlijk te verschillen van de s'aarden gevonden voor de. niet-roterende stroming. Hieruit. blijkt dat

de waarden van de coefficienten in Reynolds-spanning turbulentie modellen niet universeel

zijn maar zeerwaarschijnlijk aThangen van bet stromingsprobleem (en het Reynolds getal) dat wordt beschouwd.

Chapter 1

Introduction

1.1 Why Turbùlence?

It may seem somewhat peculiar to begin the introduction of a thesis on turbulence with the question "why turbulence?". A 'specialist' on turbulence theory will be surprised by this question and wonder "Why turbulence again? We, at least I, know everything about it ". Others, familiar with fluid dynamics in general,, might beinterested to know why we consider turbulence in particular and not one of the other interesting aspects of fluid dynamics. Finally, any layman on the area of fluid dynamics will return the appropriate

question' "What is. turbulence?". These three questioners, with their totally different

questions, have one thing in common. When reading and learning about. turbulence, they all will find out that turbulence is one of the most interesting areas of fluid dynamics and that far from everything is known about it (this makes working on turbulence even more

challenging).

Turbulence is so much part of our daily life that we are often not aware of its 'presence',

or, even benefit from it without realizing Let me illustrate this by giving a few examples of turbulent flows anyone is familiar with: the flow in the earth's atmosphere, e.g around buildings and driving cars; the flow in the pipes of your water supply system,.e.g. when taking a shower; the heated flow above the burners of your furnace; the flow inside your cup of coffee when you're stirring it firmly (in the latter example, you even benefit from the mixing property of turbulence, presumably, without realizing). Numerous examples of turbulent flows can also be given for industrial environments, like the petrochemical or aeroplane iñdustry.

Besides the practical importance of turbulent flows, we have already argued that turbulence is also challenging because it is one of the most important phenomena in nature which are still not completely understood. Several tools, like experiments and,

more recently, numerical simulations, are available to obtain information on turbulent flows in order to increase our understanding of turbulence. This thesis only reports on application ofi,umerical simulations to investigate turbulent pipe fióws, but the numerical

'A véry brief introdUction to turbulence theory, but certainly not the answerto this question, is given in section 2.1.

2 Chapter 1: Introductiòn

results are compared with experimental data as much as possible.

1.2

Numerical Simulations

Numerical simulations of turbulence are based on 'solving' the governing equations that describe the spatial and temporal evolution of flows. The background and details of the two techniques utilized in the present investigation, Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES), are giveñ in chapter 2. Such simulations provide 3-dimensional time-dependent velocity and pressure fields, that allow us not only to study the statistical properties of the flow, but also to consider instantaneous snapshots of the flow field, e.g. to investigate structures in turbulence (see Robinson (1991)). The appli-cability of DNS and LES as a numerical tool to study turbulent flows is well established now, as shown from heir numerous applications reported in the literature. We will briefly review those applications that considered wall-bounded shear-driven turbulent flows2 in

order to explain our motivation to use DNS and LES for the present study on

turbu-lent flow in a cylindrical pipe geometry. For a general review of LES and DNS

applica-tiOns, including thei specific details, we refer to review papers by Rogallo & Mom (1984),

Schumann & Friedrich (1987), Reynolds (1989), and recently by Ciofalo (1993).

One of the first LES computations of shear-driven turbulent flows is reported by

Deardorif (1970). H studied the fully developed turbulent flow between two flat parallel plates at infinitely large Reynolds number. Schumann (1973), (1975) extended Deardorif's work by incorporating a more sophisticated SubGrid-Scale (SGS) turbulence model3 that allowed for splitting of local isotropic turbulence and inhomogeneous effects near the walls. Besides the aplication of this new SGS model, Schumann also considered the flow in a concentric annu1lus by means of LES. His simulations were the first LES applications in a cylindrical geoIietry. Mom & Kim (1982), Mason & Callen (1986), Horiuti (1987), Piomelli et al. (1989), and others performed LES with various computational techniques

and SGS models (sse chapter 3), but their simulations all have in common that they

consider turbulent flows in rectangular channels.

A similar tendency is observed in recent DNS computations. Kim et al. (1987)

re-ported detailed turbulence statistics on fully developed turbulent channel flow. Compara-ble simulations wereperformed later on by Spalart (1988) (developing turbulent boundary layer along a flat plate), Lyons et al. (1991) (fully developed turbulent channel flow with heat transfer), Kristoffersen & Andersson (1993) (turbulent flow in a rotating channel) and by Gavrilakis (1992) (turbulent flow through a square duct). These simulations all apply to turbulent flows in rectangular geometries.

Recently, the treid to apply LES (and DNS) has been focussed on more complex turbu-lent flows in which eg. flow separation occurs. Typical examples are the flow over a back-ward facing step studied by Morinishi & Kobayashi (1990) and Friedrich et al. (1991), the

2Many DNS and LES applications are reported for other types of turbulent flows, such as isotropic turbulence and buoyancy-driven flows.

3A SGS model is neeled in LES because not all scalés of turbulent motion are resolved explicitly. The details of the SGS paraineteñzation are given in chapter 3, section 3.2.1.

Numerical Simulations 3

flow over a square rib across a channel by Baetke et al. (1990) and Yang & Ferziger (1991)

and the weakly separated boundary layer by Coleman & Spalart (1993) (Coleman &

Spalart report on DNS). Another example is the DNS of a backward facing step flow by Le et al. (1993) which required approximately 1100 CPU-hours on a Cray Y-MP! These sophisticated LES and DNS computations resemble the experimental observations fairly well, and illustrate that LES and DNS can be applied successfully to study flows with separation. It should be noted that the geometries considered are still rectangular.

Following the pioneering work in annuli reported by Schumann (1973), (1975), only Grötzbach (1987) considered a cylindrical annulus geometry again in which he performed combined LES and DNS computations to investigate heat transfer phenomena in liquid metals (the temperature was considered as a passive scalar, so no buoyancy effects were included). In their simulations, the ratio of radii of outer to inner cylinder varied from 5:1 (Schumann) to 4:1 and 2.1:1 (Grötzbach). The presence of the inner cylinder certainly affects the flow field, in particular when the ratio of radii is near unity. LES computa-tions investigating fully developed turbulent flow in a straight circular pipe (i.e. without inner cylinder) have been reported only recently. Apart from the present investigation (Eggels & Nieuwstadt (1991), Eggels et al. (1993)), Unger & Friedrich (1991), (1993) pre-sented their LES results for a flow in such a cylindrical geometry and obtained reasonably good agreement with experimental data.

Summarizing, it appears that LES and DNS have been applied to investigate various kinds of wall-bounded shear-driven turbulent flows, including flows which are subject to external influences, like rotation or heating. Most of these applications, however, have been restricted to rectangular geometries, whereas only a few simulations have been

re-ported in a cylindrical (annulus) geometry. From a practical point of view, flows in

cylindrical geometries are undoubtedly of comparable importance, e.g, in industrial flow problems. This view basically explains the motivation of the present work which is fo-cussed on DNS and LES of turbulent flows in a cylindrical pipe geometry. Furthermore, work in cylindrical geometries can be regarded as a prelude to LES computations in môre

general geometries.

Besides providing detailed information on flow statistics (see sections 4.2 and 4.3) and on turbulence structures (a few examples are reported briefly in section 4.2.6), LES and DNS data can also be used to check Reynolds stress turbulence models4. Computations that use such turbulence models are commonly employed to solve (complex) flow prob-lems, e.g. encoüntered in industrial processes. Alternative ways to obtain the required information are often not available. However, due to difficulties with the modelling ofthe Reynolds stresses, these computations sometimes fail to produce numerical results that are in acceptable agreement with experimental data. Typical examples of flows which are hard to handle with Reynolds stress modelling, are flows in which rotation (or swirl) plays

a significant role (Chen (1992), Parchen (1993)). One possible way to validate Reynolds stress models for such flows is based on comparison with detailed numerical data obtained

4 Chapter 1: Introduction

from DNS or LES. in orderto perform such comparison, the flow problem considered must be complex enough' to reveal the shortcomings of Reynolds stress turbulence modelling, but yet sufficiently simple to be handled by means of DNS or LES. An interesting flow problem which meets these requirements is the turbulent flow in an axially rotating pipe.

Also from a fundamental point of view, this is an interesting flow configuration, as it

already reflects some of the phenomena typical for rotating flows (see chapter 5). Before proceeding with the outline of this thesis, let us summarize the twofold goal Of the present

investigation:

Application of DNS and LES for simulations of both standard and rotating pipe

flows in order to investigate the statistical properties of such flows, and to validate conventional Reynolds stress turbulènce models,

Generation of databases of turbulent flow in a cylindrical pipe geometry in order

to support practical and fundamental turbulence research, e.g. with respect to

structures in turbulence.

To achieve this goal, several steps have to be taken. This will :be discussed in the next section where the outline of this thesis is described.

1.3

Outline of this thesis

First, the backgrouiid of Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) is considered in chapter 2. To perform such simulations in a cylindrical geometry, a computer code is developed. The details of the computational techniques involved in this code are described in chaptèr 3. In section 3.2.1, attention is paid to the SGS model required in LES. Th details of the SGS models employed in our simulations, together with the drawbacks of these models, are discussed. In the section that follows, the Reynolds stress turbulence m1odelling is briefly summarized and parts of the modelling (i.e. the strain closùre approximations) are considered in more detail. These

pressure-strain closure approximations are evaluated later on, using the DNS and LES data. In

section 3.3, the nunerical techniques utilized in our code are discussed. Attention is paid

to the spatial and temporal discretizations and to the boundary and initial conditions.

Also the difficulties with respect to the application of the cylindrical coordinate system are highlighted.

Secondly, the performance of our code is evaluated in chapter 4 by means of simulations

of fully developed urbuIent flows through straight pipes. This evaluation is based on

comparison of numerical results with experimental data. Various flow statistics obtained from DNS and LES are presented in sections 4.2 and 4.3 respectively. Subsequently, in section 4.4, we turn o the evaluation of the pressure-strain closure approximations in this

pipe flow.

Next, in chapter! 5, the turbulent flow in an axially rotating pipe is considered using LES. Several flow sttistics are reported and, where possible, compared with experimental

References

data. The effects of pipe wall rotation are investigated by comparison of the results

obtained from LES with those reported in section 4.3 for the flow without rotation. In section 5.4, a similar analysis with respect to the evaluation of the pressure-strain closure approximation as reported for the non-rotating pipe flow in section 4.4, is given for thé axially rotating flow.

Finally, the main results and conclusions are summarized in chapter 6.

References

BAETKE, F., WERNER, H. & WENGLE, H. 1990 Numerical simulation of turbulent flOw over strface-möunted obstacles with sharp edges and corners. J. Wind Eng. & md. Aerodynamics 35, 129-147.

CHEN, Q. 1992 Numerical prediction of turbulent swirling flow in a straight pipe. Internal

report MEAB-104, Deift University of Technology, The Netherlands.

CIOFALO, M. 1993 Large-eddy simulation of turbulent flow and heat transfer: A state of the art review. Dipartimento di Ingegneria Nucleare, Università di Palermo, Quaderno No. 1/93. COLEMAN, G;N. & SPALART, P.R. 1993 Direct numerical simulation of a small separation búbblé.Iñ Near-wall turbulent flows, pp. 277-286,. R.M.C. So et al. (Eds.), Elsevier, Amsterdam. DEARDÔRFF, J.W. 1970 A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41, 453-480.

EGGELS, J.G.M. & NIEUWSTADT, F.T.M. 1991 Large-eddy simulation of turbulent pipe flow. Abstracts of the ist European Fluid Mechanics Conference, Cambridge, United Kingdom, Sept. 16-21.

EGGELS, J.G.M., WESTERWEEL, J., NIEUWSTADT, F.T.M. & ADRIAN, R.J. 1993 Direct numerical simulation of turbulent pipe flow: A comparison between simulation and experiment at low Re-number. AppI. Sci. Res. 51, 319-324.

FRIEDRICH, R., AFtNAL, M. & UNGER, F. 1991 Large eddy simulation of turbulence in

engineering applications. AppI. Sci. Res. 48, 437-445.

GAVRILAKIS, 5. 1992 Numerical simulation. of low-Reynolds-number turbulent flow through a straight square duct. J. Fluid Mech. 244, 101-129.

GRÖTZBACH, G. 1987 Direct numerical and large eddy simulation of turbulent channel flows. In Encyclopedia of Fluid Mechanics, vol; 6, pp. 1337-1391, N.P. Cheremisinoff (Ed.), Gulf Pub-lishing Cömpany, Houston.

HortrnTl, K. 1987 Comparison of conservative and rotational forms in large edd simulation of turbulént channel flow. J. Comp. Phys.. 71, 343-370.

6 Chapter 1: Introduction

KIM, J., M0IN, P. & MOSER, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133-166.

KRISTOFFERSEN, R. & AÑDERSSON, H.I. 1993 Direct simulations of low Reynolds number turbulent flow in a rtating channel. J. Fluid Mech. 256, 163-197.

LE, H., MOIN, P. & KIM, J. 1993 Direct numerical simulation of turbulent flow over a

backward-facing step. Proceedings of the 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, Aug. 16-18, pp. 13-2-1 13-2-5.

LYONS, S.L., HANRATTY, T.J. & MCLAUGHLIN, J.B. 1991 Large-scale computer simulation of fully developed tubulent channel flow with heat transfer. mt. J. Num. Methods in Fluids 13,

999-1028.

MASÓN, P.J. & CALLEN, N.. 1986 On the magnitúde of the subgrid-scale eddy coefficient in large-eddy simulations of turbulent channel flow. J. Fluid Mech. 162, 439-462.

MOIN, P. & KIM, J. 1982 Numerical investigation of turbulent channel flow. J. Fluid Mech.

118, 341-377.

MomNism, Y. & }OBAYASHI, T. 1990 Large-eddy simulation of backward-facing step flow. In Engineering Tùrulence Modelling and Experiments, pp. 279-286, W. Rodi & E.N. Gaulé

(Eds.), Elsevier, New York

PARCIrEN, R.R. 1993 Decay of swirl in turbulent pipe flows. Ph.D. thesis, Eindhoven Univer-sity of Technology, The Netherlands.

PIOMELLI, U., FERZIGER, J. & MOIN, P. 1989 New approximate boundary conditions for large eddy simulations of wall-bounded flows. Phys. Fluids A 1, 1061-1068.

REYNOLDS, W.C. 1989 The potential and limitations of direct and large eddy simulations. In Whither Turbulence? Turbulence at the Crossroads, pp. 313-343, J.L: Lumley (Ed.),

Springer-Verlag, Berlin.

ROBINSON, S.K. 1991 The kinematics of turbulent boundary layer structure. NASA

TM-103859.

ROGALLO, R.S. & MOIN, P. 1984 Numerical simulation of turbulent fiòws. Ann. Rev. Flùid

Mech. 16, 99-137.

SCHUMANN, U. 1973 Ein Verfahren zur direkten numerischen Simulation turbulenter

Strömungen in Platten- und Ringspaltkanälen und über seine Anwendung zur Untersuchung von Turbulenzmodellen. Ph.D. thesis (KFK 1854), Technische Hochschule Karlsruhe, Germany. SCHUMANN, U. 1975 Sùbgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys. 18, 376-404.

References 7 SCHUMANN, U. & FRIEDRICH, R. 1987 On direct and large eddy simülation of turbulen.

Iii Advances in Turbulence, pp. 88-104, G. Comte-Bellot & J Mathieu (Eds.), Springer-Verlag,

Berlin.

SPALART, P.R 1988 Direct simulation of a turbulent boundary layer up to R9 = 1410. J.

Fluid Mech. 187, 61-98.

UNGER, F. & FRIEDRICH, R. 1991 Large eddy simulation of fully-developed turbulent pipe flow Proceedings of the 8th Symposzum on Turbulent Shear Flows, Mumch, Germany, Sept 9-11, pp. 19-3-1 - 19-3-6.

UNGER, F. & FRIEDRICH, R. 1993 Large eddy simulation of fully-developed turbulent pipe flow. In Flow sïmulation of high performance còmputérs I, NNFM, väl. 38, pp. 201-216, E.H.

Hirschel (Ed.), Vieweg-Verlag, Braunschweig.

YAÑG, K.-S. & FERZIGER, .J.H. 1991 Large eddy simulation of turbulent flow in a channel with a twodimensional obstacle. Presented at the Ist Conference on Advances in Computational Fluid Dynamics, Pohang Institute of Science & Technology, August 29-31, Pohang, Korea.

Chapter 2

Numerical Simulation of Turbulence

Abstract

In this chapter, the numerical simulation techniques which are commonly used to investigate turbulent flows are discussed. The backgrounds of Direct Nu-merical Simulation (DNS) and Large-Eddy Simulation (LES) are presented in detail. It is- emphasized -that DNS is the most rigorous numerical technique to study all details of turbulent flows. Its applicability, however, is restricted to

low-Reynolds number flows, because of the limited capacities of today 's

com-puters. LES is shown to be a good alternative in case of high Reynolds number flows. In LES, only the largest scales are computed explicitly, whereas the ef-fect of small subgrid-scales on resolved scales is modelled. It is shown that the ratio l'if, representing the range of resolved vs. subgrid-scales, is an irnpor-tant parameter in LES. Its magnitude should be sufficiently large to guarantee

a realistic LES (typically, l/l > 2).

2.1

Introduction to Turbulence Theory

In this section, several elementary aspects of turbulent flows are briefly summarized. Only the most relevant 'aspects are considered, since they will be assumed to be known further on, where. we discuss the principles of the numerical simulation techniques employed in this thesis. For a thorough introduction to turbulence theory, one is referred to a textbook on turbulence, like Batchelor (1957), Tennekes & Lumley (1972) or Hinze (1975).

A turbulent flow exhibits an irregular behaviour both in space and time. A typical

example of a time signal corresponding to a turbulent flow qUantity is shown in Fig. 2.1 where the strearnwise velocity recorded in a turbulent pipe flow is shown as fúnction of time. At first glance, the velocity may seem to behave rather randomly in time. Detàiled studies, however, have shown that turbulent flows are not cOmpletely random in space and time but that they contain spatial (coherent) structures that develop in time. These

structurés are often referred to as eddies, as they are usually associated with rotatiñg

motions of fluid flow. One fundamental result of turbulence theory is that these eddies

10 Chapter 2: Numerical Simulation of Turbulence

1.0 2.0 30

t [s]

Figure 2.1: Evolution1 of. the streamwise velocity as function of time measured in a fully developed turbulent pipe flow (by courtesy of J. den Toonder).

are not all of one particular size, but that a (broad) continuous range of large to small

eddies exists in every turbulent flow. If we return to Fig. 2.1 and carefully study the

temporal evolútion of the turbUlent signal shown there, we see that in this signal both

'fast' and 'slow' temporal variations appear which might be associated with small and

large eddies respectively. In genera], the size of the largest eddies in a turbulent flow is determined by the geometry of the flow configuration. Here, it is characterized by a length scale ¿[m]. Trpicalvalues of ¿for wall-bounded shear-driven turbulence are I O.1L with

L a length scale coresponding to the flow geometry (e.g. the pipe diameter). Besides a length scale, these large-scale eddies also. have a velocity scale denoted by u [mis]. From this, we can deduce that these large-scale eddies have a typical time scale proportional to I/u [s] and a tur1bulent kinetic energy proportional to u2 ,[m2/s2]. This kinetic energy is extracted from tue mean flow by interaction between turbulent fluctuations and mean

flow.

The smaller eddies do not extract their kinetic energy directly from the mean flow

but are fed by a continuous decay of (unstable) large eddies which break up into smaller ones; These smaller ones in turn decay to even smaller eddies, until this cascade reaches the smallest scales of turbulent motion (in. turbulence theory, this process in known as

jk

¡

I I 0.18 0.16 0.14 u)

t

>

_0.12 0.10 0.08 00

Introduction to Turbulènce Theory 11

'energy cascade'). The length and velocity scales of these smallest eddies are determined by the amount of kinetic energy transferred along the energy cascade from the large eddies towards the small eddies and by a molecular property of the fluid.

The loss of kinetic energy of the large-scale eddies is represented by the dissipation

rate e (per unit mass) which has the dimensión [m2/s3]. The dissipation rate is inde-pendent of the micro-structure' and the fluid properties since it is fully determined by

macro-structure properties only. This is expressed by the following relation which is a fundamental result in turbulence theory:

(2.1)

This relation can be interpreted as the ratio of the kinetic energy of the macro-structure

eddies (= u2) and their life-time (= i/u). Within their life-time, the large eddies lose

their kinetic energy due to a break-up into smaller eddies. As we have argued above, the micro-structure scales are not only determined by the amount of kinetic energy transfer (or, equivalently, by the dissipation rate), but also by a molecular property of the fluid. This molecular property is the kinematic viscosity z' [m2/s].

As a result of dimensional analysis, we obtain the following expressions for the length

scale , the velocity scale y and the time scale r of the smallest eddies which are known as the Kolmogorov scales:

L

i

v=(ve)

r=()2.

Since the dissipation rate is known in terms of macro-structure properties, we can easily deduce relations between the various scales of the macro- and micro-structuÈe. Substitu-tion of Eq. (2.1) into the expressions of Eq. (2.2) yields:

= Re = Re

3-

= Re (2.3) with ul

Rei =

-z, (2.2) (2.4)

which is known as the Reynolds number. For large Re1 (typical älues of Re1 used in this thesis2 are Re1 36 and Re1 210), the scales of the micro-structure become much

smallér compared to those of the macro-structure. In other words, the energy cascade

1Hereafter the large eddies are also called the macro-structure, while the small eddies are called the micro-structure.

2The values of Rez used here are still very moderate. In practical flowS, like atmospheric or industrial flows, Re1 is usually much larger with values up to i0 and higher. Generally, the Reynolds number is often expressed as Re6 = U6D/z' with U6 the mean velocity based on the flow rate and D the pipe diameter. Using ¡

1D and u - (j..

)U6, the values of Re6 used here vary from 5.400 up to 42.000.

12 Chapter 2: Numerical Simulation of Thrbulence

process dètertnines the scales of the micto-structure in such a way .that the smallest

eddies can transform their kinetic energy directly into internal energy (heat) by means of molecular viscosityJ If the kinetic energy of the macro-structure eddies is increased (i.e. Re1 becomes larger), then the scales of the micro-structure become smaller compared to i and u, in order to transform more -effectively the increased amount of kinetic energy into internal energy.

The appearance1 of a broad range of scales in a turbulent flow with the macro-structure characterized by it large eddies on one hand and the micro-structure with its small-scale eddies on the other is our point of departute to illustrate the principles of the numerical simulation techniques employed in this thesis.

2.2

Principles of Numerical Simulations

2.2.1

Introduction

The equations that describe the spatial and temporal evolution of a fluid flow have been known for a long time. These equations consist of conservation laws for mass, momen--turn and energy where the energy equation is usually reformulated in terms of the fluid temperature. In this thesis, we will only consider isothermal flows, hence the energy (or temperature) equation will not be used. For an incompressible, -Newtonian, fluid -flow conservatibn of mass and momentum are expressed in vector notation by the following

relations:

0v 1 ₂

+v.Vv=--Vp±vVv

(2.6)

with t, the velocity vector, p the fluid density, p the pressure and u the kinematic viscosity. The Reynolds number introduced in the previous section can be interpreted as the ratio

of the advection term (y . Vv) and the viscous term (uV2v), if these terms are scaled

using the macro-structure scales:

''

12

--- Rej (2.7)

For large Rej, the nonlinear advection term dominates over the viscous term and, in

general, the flow will be turbulent. The nonlinearity of the advection term is the reason for the appearance of a broad range of (length) scales in turbulent flows. If the viscous

term dommates over the advection term (i e Rej is small), the flow is laminar and

characterized by regular flpw motiOn.

Since the equations which describe the flow field are known, it should be possible in principle to solve them in a discretized form by means of a (powerful) computer. Such a

Principles of Numerical Simulations 13

numerical imulation should resolve the spatial and temporal evolution of the flOw field in all detáil to capture all relevant flow phenomena. For laminar flow with small Re1, the equations ofmotion (Eqs. (2.5) and (2.6)) can be discretized and solved straightforwardly from a computational point of view. For turbulent flows, the situation, is different and

more complicated. In the remainder of this section, we will point out the principles of

the numerical simulation techniques for turbulent flows employed in this thesis. We will

not yet go into the details of the computational techniques, since that will be done in

chapter 3.

2.2.2

Direct Numerical Simulation

Numerical solution of the differential equatiOns (2.5) and (2.6) involves approximations to obtain these equations suitable for a computational approach. In general, this means that the differential equations are discretized and that space and time are represented by a network (or grid) of discrete points. Let the distance between two sequential points in

space and time be denoted by ¿x and /t respectively. As we have seen in the previous

section, turbulent flows are characterized by a broad range of length and time scales that

should be resolved in all detail by the numerical simulation. Hence, zx and Lt should be proportional to the smallest length and time scales respectively. In other words, to

perform a realistic numerical simulation of a turbulent flow, the Kolmogorov length and time scales should be resolved by the spatial and temporal grid distribution from which follows that x

ri and /t

T (these criteria3 hold for homogeneous isotropic turbulent flows). However, a more restrictive criterion must sometimes be applied for t in which Ta with Ta, a time scale proportional to the time required for a small-scale eddy to pass a fixed point when advected by the macro-structure velocity, i.e. a time scale

proportional to ri/u (Nieuwstadt (1993)). A numerical simulation in which all relevant scales of turbulent motion are resolved explicitly is called Direct Numerical Simulation (DNS) or Fully Resolved Simulation (FRS).

The relations between the macro- and micro-structure scales are given by Eq (2.3). Using I '-' O.1L, it follows':

k

= 1ORe,. (2.8)

For the ratio of T and ¿t with T a time interval during which the flow field is monitored, a similar expression can be obtained. As for the length scales i and L, we have to relate

T to the time scale I/u of the macro-structure. To obtain a correct impression of the

turbulent flow field, the flow should be monitored for at least several time scales I/u. In

our analyses to collect the results presented later on in this thesis, we used T '- 501/u.

Then we find the following expression: T 501/u

50Re (2.9)

3For wall-bounded shear-driven turbulent flows, alternative criteria are also used. They vill be con-sidered further on.

14 Chapter 2 Numerical Simulàtion of Tutbulence

in which we used the most restrictive criterion for &. The ratio T/Lt is denoted by

NT and represents the number of time steps which have to be taken in order to resolve

all relevant time sales of turbulent motion during the interval T. Similarly, the ratio

L/Lx is denoted by NL which represents the number of gridpoints needed to resolve all relevant length scales of turbulent motion in one direction. Since turbulence is essentially three-dimensional, the numerical simulations must also be three-dimensional and so the

total number of gridpoints is proportional to N.

Eqs. (2.8) and (2.9) hold for direct simulations of turbulent flows in which the Kol-mogorov scales ar the characteristic scales. For wall-bounded shear-driven turbulent flows, an alternative scaling of the smallest eddies is also used (see Reynolds (1989)). Here, the thickness of the viscous wall layer, which scales like v/u, is assumed to deter-mine the smallest scales throughout the flow. As a result,

x --' v/u and tt

v/u2 for the same reason as previously for Ta. Following the same analysis as above, we now end up with:

M

T

in which the powers to which Re1 is raised hâve become equal tounity. Hence, the require-ments on spatial and temporal resolutions have become stronger compared to Eqs. (2.8) and (2.9). To justify the numerical resolution used for the DNS of turbulent pipe flow

presented in chapter 4, both scalings of the smallest eddies are considered (using the

Kolmogorov length scale as well as the thickness of the viscous wall layer). For the con-tinuation of the present discussion, it is not directly relevant what type of scaling is used (the final conclusion will be similar). So, we will continue with the scaling based on the

Kolmogorov scales.

Recently, Nieuwstadt et al. (1993) estimated the required computer memory M (in bytes) and the computer time T (in CPU-seconds) based on the parameters introduced above in order to perform a realistic DNS. He showed that:

= 8nCMN = 8 l0neÇMRe

(2.12)

=

T1eNOP3NN3 _{= . (2.13)}

'1ff

'it!

In these relations, e denotes the number of. flow quantities to be solved. CM and N3

are two parameters depending on the computational technique involved in the DNS.

The peak performance of the computer system in floating point operations per second (flops) is given by f whereas r is an efficiency parameter indicating which fraction of the

peak performance is achieved in the actual computation (the latter parameter strongly depends on the attainable degree of vectorization of the code). The values of n and f are respectively given by the physical problem considered and the computer system

NL =

lORe1 (2.10)

T

Principles of Numerical Simulations 15

used. The other parameters are estimated from actual DNS computations. Following Nieuwstadt et al. (1993), we use n = 4, CM = 3.1, N8 = 200, f = 333 [Mflops] and

= 0.4 which yields after substitution4:

5 12

IyI = 105Re and T = 0.3Re,4.

Both M and T are proportional to Re, raised to a positive power larger than unity. With increasing Re,, M and T thus increase rapidly. The DNS computation describeä in section 4.2 of this thesis was performed at Re, = 36 (Ree = 360). Using Eq. (2.14),

we end up with M = 317 Mbyte] and T = 1.4 iO

CPU-seconds]5. This is still

within reach of today's superÇomputers, only because of Re, being low. It is obvious

that with increasing Re, (say, Re, up to i0), DNS computations cannot be performed

anymore because of the limited capacities of today's supercomputers. Nevertheless, for small Re,, DNS is the most favourable numerical simulation technique to study all aspects of turbulent flows because it is based on direct solving of the governing equations without any additional assumptions.

2.2.3

Large-Eddy Simulation

A remedy to overcome the limitations of DNS with increasing Re, is to reduce the range

of scales that are resolved on the numerical grid. A possible concept is to remove the

small-scale eddies by a spatial filtering procedure and to resolve the large-scale eddies only. This approach is called Large-Eddy Simulation (LES). The separation of small and large scales is inspired by observations that the large eddies of the macro-structure are mostly anisotropic. Furthermore, they depend on the geometry of the flow considered. On the other hand, the small eddies of the micro-structure can be considered to be closer to isotropic. They are much less dependent on the flow geometry, as they are fed by the energy cascade in which the geometry information present in the larger eddies gets lost due to break-ups. Therefore, the micro-structure eddies may perhaps be regarded more or less universal. Since in LES the large eddies are resolved explicitly on the spatial grid (similar as is done in DNS for all scales of turbulent motion), only the effect of removed small scales remains to be modelled. The (more) isotropic and (perhaps) universal behaviour of the small scales is a favourable point of departure for the modelling of the small-scale eddies. This modelling is referred to as subgrid-scale (SGS) modelling and we will return to it in section 3.2.1.

Let us now consider the equations that need to be solved in LES. In LES, each flow

variable is decomposed into a large-scale (or grid-scale (GS)) component and a

subgrid-(2.14)

4The precise magnitudes of the preceding coefficients are not very important yet because at the momént we are mainly interested in the dependence on Re,.

5The computation time T in. the actual DNS computation reported in this thesis is much larger than follows from Eq. (2.14) because in this computation the time step tt is not determined by the time scale Ta but by restrictions to ensure numerical stability. As a result, the admissible time step t is

múch smaller (factor 15!) than the time scale Ta and so the number of time steps NT, and hence the computational time, is strongly increased (for further details refer to section 3.3.2 and 4.2).

16 Chapter 2 Numerical Simulation of Turbulence

scale (SGS) component Ø' as 4 = + '. The GS component is defined by the

moving-average ifiter operation: (z, t)

=

f G(z - z(z', t)dz'

(2.15)

in which G(z - z') is a spatial filter function depending on the separation between the

spatial vectors z and z'. V is the volume df the computational domain. The filter function is often chosen to be Gaussiàn or top-hat (Leonard (1974), Piomelli et ai. (1988)) where,

in case of the top-iat distribution, G(z z') is given by

-G(z

z')

{J

O elsewhere

with a characteristic filterlength. Application of the spatial filter operation (2.15) to Eqs. (2.5) and (2.6) yields the equations for the GS velocity vector Y:

(2.17)

(2.18)

at

p

where the continuily equation (2.17) is used in the momentum equation (2.18) to

refor-mulate the advectie term into its conservative form. The stress tensor i represents the

influence of the SGS turbulent motions on the GS velocities and is given by:

(2.19)

In section 3.21, the various terms in Eq. (2.19) will be discussed. For the moment, it is only important to note that r includes the SOS component y' of the velocity vector y and hence is unknown. Expressions for r need to be formulated either by deriving prognostic partial differential equations for all its components, or by relating r in än algebraic way

di±ectly to the OS velocity vector i. For details regarding the SGS modelling of r, we

refer to section 3.2.1. Here, we will continue by looking at the computer requirements in case of LES, similar as done before for DNS.

In LES, the macro-structure with its length scale lis still resolved on the spatial grid. The filterlength i of the applied filter function G(z - z') is now a characteristic length

scale of the smallest GS motions as well äs of the largest SGS motions. For LES to be realistic, the ratio l/l representing the range oflength scales of the resolved GS motions, should satisfy l'if > 1. If l/i < 1, then even the large-scale eddies of the macro-structure are removed by the spatial filtering which means that all turbulent length scales become

part of the SGS motions and the fundamental idea of LES gets lost6. The filterlength

if Iz'zIeV with

(2.16)

6 _{general, a similar type of filtering with l/1 «i is applied in Reynolds stress turbulence models.}

N

L==Ti2Oi

L

Liif

Principles of Numerical Simulations .17 i can be related to the gridspacing where one usually assumes 1

.= 2x (Nyquist

criterion; see also Mason & Callen (1986)). Larger values of i, with unchanged ix (i.e. 11/&r> 2) yield a more accurate numerical representation of the smallest resolved scales but simultaneously reduce the range of resolved GS motions without any reduction of the computational effort. In LES, one. usually aims to keep the range of GS motions as large as possible and therefore retains to 2zx. Thus, we can define the ratio NL now as:

In contrast to Eq. (2.8) for the DNS, NL is not related to Re1 anymore but to the ratio

i/Ij. As a result, LES is not restricted to low Re1 values only but can also be applied for simulations of turbulent flows with high values of Re1. The coefficient '20' in Eq. (2.20)

appears due to the product of the parameters L/i and if//.x. It has already been argued

that the parameter L/i depends on the type of flow considered and for shear-driven tur-bulent flows considered in this thesis, it follows Lii 10. For buoyancy-driven turbulent flows (e.g. the convective atmospheric boundary layer), the ratio L/i is close to unity and hence gives much lower values for NL at comparable i/is. In other words, largér values of i/is can be applied in LES of such turbulent flows to end up with óomparable values of NL as fOr LES of shear-driven flows. The second parameter, i/&, describes the ratio of filterlength to gridspacing and, as we have argued above, should preferably be taken close to 2.

From the previous paragraph, it appears that the ratio i/ij' plays a central role in LES

computations. Let s now look somewhat closer to the significance of this parameter. Large values of i/is (say, i/if> 10) correspond to realistic LES computations, in which a broad range of turbulent length scales are resolved explicitly and in which a limited range of length scales are incorporated into the SGS stress tensor r. Typical examples of such LES applications are simulations of the convective atmospheric boundary layer. Large

values of 1/ii are possible here because the ratio L/1 is small in such flows. Nevertheless, the demand on computer power (memory and CPU time) will increase rapidly with in-creasing i/i1, as it immediately determines the spatial resolution NL. On the other hand, LES computations with small values of i/Is (say, i/if <5) might be considered somewhat less realistic. Here, the range of resolved GS motions is small and the dependence on the SGS closure model might become more important. Nearly all simulations of (wall-bounded) shear-driven turbulent flows, including those reported in this thesis, correspond

to the latter category of LES computations as a result of L/i being large, which forces i/if to be limited.

Let us finally turn to the temporal resolution NT in LES computations. We have seen that the smallest resolved motions in LES are characterized by the filterlength i.

On similar grounds as before for the DNS, the time step t should thus be proportional

to the time scale if/U. In general, however, ¿t in LES computations is determined by

criteria to ensure numerical stability of the computation rather than by other criteria (see section 3.3.2). Assuming that the Courant criterion (refer to appendix A) is the relevant

criterion here for the determination of t, we obtain:

18 Chapter 2: Númerical Simulation of Turbulence

C:=

=

(2.21)

with CI < i denoting the Courant number and a the ratio of the relevant advection

velocity to the macro-structure velocity scale u ( is of the order 20 to 40). The ratio of

Lt and lj/u can easily be obtained now as:

Lat CLXx

= ---= «1

lj/u

a

i

which is always much less than unity. Hence, the time step t imposed by the stability criterion is always sùfficiently small compared to the time scale 11/u.

2.2.4

Reynolds-averaged Modelling

To complete our survey of computational techniques to investigate turbulent flows, we will briefly consider Reynolds-averaged Modelling (RaM) here. Time-averaging is employed in RaM to reduce the range of scales present in a turbulent flow. The averaging time is much larger than the largest time scale of the turbulence fluctuations and as a result, one ends up with equations f motion that describe the evolution of the mean flow quantities. only. The influence of the removed turbulence fluctuations on the mean flow is incorporated iñto the. so-called Reynolds stress tensor which is the Reynolds-averaged 'equivalent' of

the SGS stress teisor r. One of the differences between LES and Reynolds-averaged models is the range of turbulent scales embedded in its stress tensor components. In

Reynolds-averaged models, the Reynolds stress tensor describes the influence of all scales of turbulent motion (including the anisotropic large scales) on the mean flow whereas in LES the SGS stress tensor only reflects the influence of small-scale turbulence on GS flow

quantities. Due to the inclusion of large-scale turbulent motions into its stress tensor

components, the modelling involved in RaM appears to be fairly complicated (see e.g. Launder et al. (1975), Hanjalié & Launder (1976), Launder (1989a), Parchen (1993) and many others). It is beyond the scope of this thesis to describe in detail all aspects of

Reynolds stress modelling. Nevertheless, in section 3.2.2 the major aspects of this type of turbulence modelling will be summarized briefly, as they will be used further on in this

thesis.

(2.22)

References

BATCHELOR, G.K. 1957 Homogeneous Turbulence. Cambridge University Press.

HANJALIÓ, K. & LAUNDER., B.E. 1976 Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence. J. Fluid Mech. 74, 593-610.