Routes to Absolute Instability in Porous Media

289 Pages·2019·5.51 MB·English

by  Antonio Barletta

Antonio Barletta
Routes to
Absolute
Instability in
Porous Media
Routes to Absolute Instability in Porous Media
Antonio Barletta
Routes to Absolute Instability
in Porous Media
123
AntonioBarletta
Department ofIndustrial Engineering
AlmaMater Studiorum Università
di Bologna
Bologna, Italy
ISBN978-3-030-06193-7 ISBN978-3-030-06194-4 (eBook)
https://doi.org/10.1007/978-3-030-06194-4
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What we cannot speak about we must pass
over in silence.
Ludwig Wittgenstein
Foreword
Stabilitytheoryhasan oldandvenerablehistory inthecontextoffluidmechanics:
much is known and much remains to be discovered. In our desire to model accu-
ratelyourhighlynonlinearworld,itisquitenaturaltobeginslowlybydetermining
first the flow field for small values of a parameter such as the Reynolds, Rayleigh,
TaylororGörtlernumbers.Insuchcases,theflowisslowandnonlineareffectsare
barely felt. Perhaps it is unsurprising then that such flows are stable, not that we
have yet defined this word. One potential definition of stability might be that the
flow is unique, but this idea is unsatisfactory because it mentions neither the
presenceofnortherôleplayedbydisturbancesofanykind.Considerationofthese
matters is central to stability theory and is essential before pressing on to fully
numerical simulations and the transition to turbulence.
We all have an intuitive idea of what the words stable and unstable mean
because they may be related to many aspects of human life and experience:
nitroglycerine is unstable; a cyclist in motion is stable; a poorly constructed
buildingisunstable;apersonmaybesaidtohaveastablecharacter.Eachofthese
examples gives a hint as to what stability might mean in the context of fluid
mechanics. A small disturbance of the form of an impact applied to nitroglycerine
willcauseittoexplode.Evenapoorlyconstructedbuildingwillcollapseonlywhen
the magnitude of an earthquake exceeds a certain value. An experienced cyclist
who has spent a lifetime successfully negotiating difficult terrain may be desta-
bilised (i.e. floored) by the sudden but unwelcome presence of a squirrel in the
wrong place. People with highly stable characters are able to manage to cope with
theusualpressuresoflife,butwhenalargedisturbancearises(Iamthinkinghereof
chronicjobstresses,domesticissues,abereavement,amuggingandsoon),thena
breakdown of some kind could follow and life changes subsequently. All four
oftheseexamplesmaybeinterpretedintermsofhowthenatureandthemagnitude
of the disturbances can alter forever the original state.
Returning to fluid mechanics, it is well known that some flows, which we may
call basic states, will be destabilised by the presence of infinitesimally sized dis-
turbances,andthatthevalueofthegoverningparameterabovewhichthishappens
may be minimised by the appropriate selection of the shape of the disturbance.
vii
viii Foreword
Theanalysiswhichdeterminesthecriticalvalueoftheparameteristermedalinear
stability theory since the temporal growth of the disturbance satisfies a linear
evolution equation at least initially. A growing small-amplitude disturbance will
eventually saturate; the new flow will have a different appearance from that of the
basic state, and it also has fewer symmetries. On the other hand, other basic states
may resist small disturbances at a given value of the governing parameter but are
neverthelesshelplesstoresisttheeffectoflargerones.This,then,istherealmofthe
energystabilityanalysiswhichmayalsobeusedtodeterminethecriticalparameter
below which nonlinear flow does not exist. Weakly nonlinear theory may also be
used to further the understanding of the full behind-the-scene behaviour of the
system being studied, and this includes pattern selection and bifurcations.
These different types of analysis, followed by comprehensive nonlinear simu-
lationsofthegoverningequations,formthebackboneofstabilitytheoryingeneral.
Theyhavebeendescribedindetailbothinmonographsandinabewilderinglylarge
numberofjournalpapersoverthelast100yearsorso.Thisremainstrueevenwhen
one restricts attention to convection in porous media. The fact that different basic
flows have a multitude of potentially different routes towards turbulence makes
stabilitytheoryaveryinteresting,challengingandrewardingtopictostudyevenif
one does not consider its utility.
The present book is devoted to the concepts of absolute instability and con-
vective instability, a topic which is very new in the field of porous medium con-
vection. When a flow is absolutely unstable, a disturbance placed in one locality
will continue to grow within that locality. This does not preclude it spreading or
diffusing into formerly undisturbed regions as time passes. On the other hand, a
convective instability will correspond to a disturbance which also grows in time,
but where the background flow field is sufficiently strong that the disturbance will
eventually decay within the region where it was introduced. In other words, the
growing disturbance leaves the scene of the crime! The word, convective, when
used in this way, refers to the carrying away of the disturbance, an idea which
reflects the Latin etymology of the word. However, there is a potential confusion
between the use of the term, convective instability, to describe such a moving
instability mechanism and its use when talking about systems such as Bénard
convection which is a buoyancy-induced instability. Generally, the context will be
unambiguous, but the careful author might need to state something like the fol-
lowing in order to guarantee clarity: the present thermoconvective instability is a
convective instability as opposed to being an absolute instability.
Professor Barletta has crafted a monograph which describes with great clarity
howtodeterminewhetheraninstabilityisabsoluteorconvective.Todothis,hehas
set the scene in the earlier chapters by introducing the reader to the Fourier and
Laplacetransformsand,inparticular,totheirroleinthestudyoftheevolutionand
movementofwavepackets.Theseareillustratedusingsimplifiedsystemsofpartial
differential equations which display unstable behaviour of the kind which shows a
transition between convective instability and absolute instability. Part II brings the
reader an introduction to flows and convection in porous media, and a whole
chapter is devoted to the porous medium analogue of the Rayleigh–Bénard
Foreword ix
problem, often called the Darcy–Bénard problem or the Horton–Rogers–Lapwood
problem. Then, in Part III, the Prats variant of the Darcy–Bénard problem is con-
sideredwhereanexternallyappliedpressuregradientdrivesahorizontalbasicflow.
This is analysed in detail using all the theoretical concepts developed earlier, and
thetransitionfromconvectivetoabsoluteinstabilityisstudied.Theanalysisisalso
extended to the equivalent vertical layer with sidewall heating and with fixed
pressureprofilesatthesurfaces.Thefinalchapterandtwoappendicescompletethe
work, and these describe ancillary matters.
Thepresentbookfillsanimportantgapinthemarketbecauseithasbeenwritten
specifically to introduce the reader to the relatively recent topic of convective and
absolute instabilities but within the context of convection in porous media where
these ideas are unknown to all but a select few.
Finally,Iwouldliketosayafewwordsabouttheauthor.Icounthimasavery
good friend indeed and occasional confidante. Antonio and I have collaborated in
the publication of many journal and conference papers in the last 10 years, and I
look forward to very many more. The present book represents, in a way, a distil-
lationofanalternativelifeofhis,onethatIdonotknowandhavenotbeenpartof,
where he has collaborated with others, and I have thoroughly enjoyed reading it
fromendtoend.Inowknowmuchmoreabouttheangstthatthewritingofabook
caninduce,butthisexperiencehasnotdestabilised him!Rather,thebookdisplays
well his clear thinking, erudite style andteachingability. Ihave learnedsome new
tricks, and I hope that very many others do the same.
Bath, UK D. Andrew S. Rees
October 2018 University of Bath
Preface
It is an impression of this author that there is a disproportion between the plethora
ofstudiesregardingtheconvectiveinstabilityinporousmediaandthosewherethe
analysis is pushed forward to investigate the transition to absolute instability. The
possible reason is that this concept is hardly recognised among the shared
knowledgewithintheporousmediacommunity.Thisisthemainmotivationbehind
this book. In fact, this book is aimed to provide all the information necessary for
approaching an analysis of absolute instability for applications to flow and heat
transfer in fluid-saturated porous media.
Thisbookisforthosewhostudyconvectionheattransferatagraduateleveland
have a specific interest for porous media applications. Another possible target is
thosewhoarefamiliarwiththetopicofmechanicalinstabilityandaimtoenterthe
research topic offluid mechanics and heat transfer in porous media. In both cases,
this book is meant to be a pedagogical guide that can be employed in a class on
thermal instability in porous media, or even as a self-learning tool.
Allthemathematicsneededtounderstandthetopicsofconvectiveandabsolute
instabilities is provided in Part I of this book. Chapters 2 and 3 provide a
self-contained introduction to the basic mathematical methods employed for the
assessmentofinstabilityinflowsystems,namelytheFouriertransformandcalculus
with complex variables. Chapter 4 introduces the concept of instability in a
mechanical system with a finite number of degrees of freedom extending this
concepttoacontinuoussystemwhoseevolutionisdescribedbyapartialdifferential
equation. This chapter provides formal definitions of convective and absolute
instabilities.
Anintroductiontotheessentialelementsoffluidmechanicsandconvectionheat
transfer is provided in Part II, where Chap. 5 contains the description of the
mathematical model of fluid flow, Chap. 6 discusses the basic ideas behind the
theory of fluid mechanics in porous media, while Chap. 7 contains a discussion
of the thermal instability through the classical case study of the Rayleigh–Bénard
problem and its variants. Among them, the applications to saturated porous media
are investigated, starting from the Horton–Rogers–Lapwood problem.
xi
xii Preface
Part III has its focus on the transition from convective to absolute instability in
porousmedia.Chapter8containsadiscussionofPratsproblemincludingavariant
formulation where the effects of the form-drag effect are taken into account. In
Chap. 9, cases where the analysis of convective and absolute instabilities is
approached numerically are presented. Finally, Chap. 10 offers a detailed
descriptionofanumericalmethodforthesolutionofstabilityeigenvalueproblems
for the convective or the absolute instability.
What is not present in this book is the nonlinear approach to instability. This
choice has been made to keep the mathematical difficulties at their lowest and
becausethelinearapproachisinitselfsufficientlywideanddiversifiedastoprovide
a very large amount of information. More than aiming for completeness in the
presentationofconvectiveandabsoluteinstabilities,ithasbeenchosentofollowthe
pedagogicalway:discussless,butbeasdetailedandcomprehensibleasyoucanbe.
Bologna, Italy Antonio Barletta
October 2018