DNS of Wall-Bounded Turbulent Flows

393 Pages·2019·60.546 MB·English

by  Tapan K. Sengupta, Swagata Bhaumik

Tapan K. Sengupta · Swagata Bhaumik
DNS of Wall-
Bounded
Turbulent
Flows
A First Principle Approach
DNS of Wall-Bounded Turbulent Flows
Tapan K. Sengupta Swagata Bhaumik
(cid:129)
DNS of Wall-Bounded
Turbulent Flows
A First Principle Approach
123
Tapan K.Sengupta Swagata Bhaumik
Department ofAerospace Engineering Department ofMechanical Engineering
Indian Institute of Technology Kanpur Indian Institute of Technology Jammu
Kanpur,Uttar Pradesh Jammu,Jammu &Kashmir
India India
ISBN978-981-13-0037-0 ISBN978-981-13-0038-7 (eBook)
https://doi.org/10.1007/978-981-13-0038-7
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Preface
A long habit of not thinking a thing is wrong gives it a
superficial appearance of being right. Reason obeys itself,
andignorancesubmitstowhateverisdictatedtoit.
TomPaine
There are many books and monographs on instability and transition, including
onebyoneoftheauthorsofthismonograph.Despitethesebooks,thismonograph
is written with a specific purpose of presenting a detailed account of what consti-
tutes direct numerical simulation (DNS) of flow from receptivity stage to fully
developed turbulent flow over flat plate, while simulating the classical transition
experimentsforzeropressuregradientboundarylayer.Thishasbeenmadepossible
with the work of the authors’ colleagues and students, proposing new theoretical
andcomputationalresults,whichestablishadeterministicroutetoturbulence,asin
the classical experimental efforts [3, 5, 7].
TheNavier–StokesequationgovernsfluidflowsandReynoldsexplainedthatthe
exact solution of the Navier–Stokes equation, even when it exists, is unable to
maintainitsstabilitywithrespecttoomnipresentsmalldisturbancesintheflow.An
equilibrium solution ensures satisfaction of conservation principles of mass,
momentum,andenergybalance,yettheflowinthefamouspipeflowexperimentof
Reynolds disintegrated into sinuous motion, eventually leading to turbulent flow.
Thisonsethasbeenattributedtotheinstabilityofflowseversince.However,even
today the transition to turbulence in pipe flow is not completely understood.
Another canonical problem which attracted the attention of researchers in the
beginning is the flow over a flat plate. In studying instability of this flow field,
Rayleigh developed his stability equation, but it was not amenable to study con-
vective disturbances, which is the signature of transition for zero pressure gradient
boundarylayerflow.Bythebeginningoftwentiethcentury,itbecameevidentthat
the instability problem is intractable without the inclusion of the effects of second
derivative of velocity profile and viscous actions. These latter effects have been
termedasresistiveinstability,whichisincontradictiontotheassumptionmadeby
v
vi Preface
earlypioneers,whomistakenlyconsideredtheactionofviscousforcestoattenuate
disturbances and was considered not central to the study of instability.
ThiswasthereasonforOrrandSommerfeldtoindependentlycomeupwiththe
famous Orr–Sommerfeld equation, which was investigated by Tietjens [14] and
Heisenberg [4]. However, it was the definitive attempt by Tollmien that paved the
understanding the concept of critical Reynolds number [15]. Following this lead,
Schlichting also studied the instability of zero pressure gradient boundary layer,
while making some implicit assumption which connected the temporal and spatial
growthofdisturbances,whichhassincebeenaddressedbyinvokingtheconceptof
group velocity [8]. One of the central themes of Tollmien and Schlichting’s work
was to show the presence of growing waves in limited part of the flow, and this is
now called the Tollmien–Schlichting (TS) waves. This eigenvalue analysis is
performed by neglecting the growth of boundary layer. We would like to mention
thatstudyingflowinstabilitybytemporalandspatialtheoryandartificiallypatching
thetwotogetherisfraughtwithdanger.Thishasbeendemonstratedbytheauthors
and colleagues that there is really no need to make this artificial demarcation into
spatialandtemporalroutes.Thebestcourseofactionistoadoptthespatio-temporal
route to avoid this ambiguity. While studying instability of mixed convection
boundary layer, authors came across a perplexing situation, when a singularly
cooledplatedemonstratedsimultaneouspresenceoftemporalandspatialinstability
[10].TheproblemwasresolvedbecauseoftheavailabilityofDNSresultsobtained
by high-accuracy method, which showed the flow to follow primarily temporal
growth, and not the spatial route (the growth should be strictly stated to be via
spatio-temporal route). This emphasizes the need to perform spatio-temporal
analysis, which has been practiced and yielded new insights, which form the
contents of this treatise.
However, despite Taylor demonstrating the strength of linear stability of flow
insideconcentriccylinder[12],TSwavecouldnotbedetectedexperimentally.This
created doubts about the existence of TS wave and, in turn, on the relevance of
viscous linear theory. This problem was resolved by Dryden’s group, by demon-
strating for the first time the existence ofTS wave by vibrating a ribbon inside the
boundarylayer,withtheresultsannouncedaftertheSecondWorldWar[3,5].This
experimentalapproachmaybetermedasfrequencyresponseofthefluiddynamical
system.Itwouldbeappropriatetonotethatexcitingtheflowatanyfrequencywill
notleadtoflowinstability,asthevibratingdiaphragmexperimentofTaylor[13]at
2 Hz demonstrated. The frequencies to be excited were present in the works of
Tollmien and Schlichting [8, 15], but Taylor’s experimental vibration was at too
low a frequency [13]. This aspect of how to experimentally study eigenvalue
problemisnotclearlyunderstood.Inarecentwork,researchershaveexplainedthe
relationship between frequency response and instability for an associated problem
of bifurcation [6].
Viewing fluid flows as dynamical systems, it is thus unwise to ignore the
receptivity aspect. Receptivity was emphasized by Schubauer and Skramstad in
reporting their classic experimental results [7]. Thereafter, reported TS waves as
Preface vii
earlymarkerofflowinstabilitywasreadilyembracedbytheresearchcommunityto
conclude prematurely that TS waves are the cause for flow transition. It is worth-
while remembering that vortical excitation validated instability theory, while the
acoustic excitation did not! Also, instability theory does not require any specific
excitation,exceptinprescribingthequalitativenatureofboundaryconditions.The
main feature of this monograph is to relate receptivity with flow instability and
show how different routes of excitation lead to different types of disturbance
evolution. This has been achieved here primarily by DNS to explain theoretical
aspects of the flow.
Viscous instability results also suffered credibility due to the use of a parallel
flow assumption for the equilibrium flow, along with linearization. This criticism
waspartlysilencedbyexperimentalverificationprovided[7],butwhetherTSwaves
cancausetransitionornotwasnotknownforalongtime,tillthedefinitiveroutesof
transitionwereidentifiedfor2Dzeropressuregradientflow[9].Ithasbeenshown
by the present authors that when the Navier–Stokes equation is computed in 2D
framework to mimic the experiment [7] for moderate- to high-frequency wall
excitation, created TS wave packet remains passive, while the spatio-temporal
wave-front (STWF) causes transition. The STWF is the first wave front that is
createduponstartingofftheexcitation,asinanexperiment,whichhastheproperty
to regenerate other STWF in its wake [9]. It is interesting to note that STWF was
originally proposed in research on electromagnetic wave propagation [2]. In fluid
dynamics,themajorsuccessintrackingSTWFfromthespatio-temporaldynamics
ofOrr–Sommerfeldequationwasreported[11],followingthedevelopedtechnique
of Bromwich contour integral method. While these have been obtained for the
frequency response of disturbance evolution, corresponding role of STWF during
impulse response has been reported only recently [1], which also shows that the
same physical mechanism of STWF explains the features of geophysical phenom-
ena like tsunami and rogue waves.
This book has evolved into an account of the research interests of the authors
over the years. Efforts are made to keep the treatment at an elementary level
requiring rudimentary knowledge of calculus, Laplace–Fourier transform, and
complexanalysis,whichshouldbeequallyamenabletoundergraduatestudents,as
well as serious researchers in the field of hydrodynamics and mixed convection.
Thismonographshowstoreadersthatwithoutgoodcomputing,thissubjectwillbe
poorer in linking spatio-temporal growth, instability at low frequencies, and the
actual physical mechanisms of transition. In providing computational results from
receptivitytoafullydevelopedturbulentstageof2Dand3Ddisturbanceflows,we
also definitely provide the basis of experimental approach for transition to turbu-
lence. In doing so, we state that turbulence is deterministic in its origin, as it is
implicitly assumed in transition experiments [3, 5, 7]. Trained with high-accuracy
computing methods, users of this monograph will be able to further contribute in
this rich field of nonlinear dynamics.
Thus, the emphasis of the monograph is on DNS of transitional and turbulent
flows,basisofwhicharelaidoutinChap.2.TheapplicationsofDNStechniqueare
usedtoexplainreceptivityandinstabilityinhydrodynamics andmixedconvection
viii Preface
flows in Chap. 3. The power of DNS is demonstrated by using results to explain
vortex-induced instability in a nonlinear framework to explain receptivity and
instability in Chap. 4. This is done by developed disturbance enstrophy transport
equation [16, 17]. This provides a nonlinear framework based on Navier–Stokes
equation without making any assumption on the equilibrium flow. Another non-
linear framework to study instability in this chapter is by using enstrophy-based
proper orthogonal decomposition. The last two chapters are to demonstrate the
powerandaccuracypresentedbytheDNStechniques described inearlierchapters
toshowthatSTWFistheprecursorsoftransitiontoturbulence,forboth2Dand3D
disturbanceflowsimulations.Theresultingturbulentflowsareashavebeencreated
experimentally[3,5,7].Thetransitiontoturbulenceisshowntobeexperimentally
obtained for the integral properties and presented in textbooks. In closing this
discussion, we note that the subject has matured very rapidly in recent times with
the advent of very high-accuracy methods of computing, which will lead to an
explosive growth of activities in the subject field. The contents of the monograph
canbeadoptedastextforahigh-levelcourseonDNSandtransitiontoturbulence.
The contents of this monograph are based on the doctoral thesis work of one
of the authors and the other graduate students who have been associated with the
firstauthoroverthelastthreedecades.Thecontentshavebeenspecificallyenriched
by faculty colleagues in NUS, Singapore, specifically by Profs. Y. T. Chew,
T.T.Lim,B.C.Khoo,K.S.Yeo,andS.ChangandProf.PierreSagaut(UPMC),
Prof. Julio Soria (Monash University), Prof. W. Schneider (TUW, Vienna), Prof.
K. R. Sreenivasan (ICTP and NYU), Prof. M. Klocker (University of Stuttgart),
Dr.J.M.Kendall(JetPropulsionLab,USA),Dr.B.R.Noack(Pprime,University
of Poitier, France), Prof. M. Deville and Prof. F. Gallaire (EPFL, Switzerland),
Prof.S.Girimaji(TAMU,USA),Prof.P.J.Strykowski(UniversityofMinnesota),
and Prof. A. Tumin (University of Arizona). Much of our work on receptivity has
been influenced by the works of Prof. H. Fasel (University of Arizona). The first
author acknowledges the influence of Prof. M. Gaster and Prof. D. G. Crighton
(University of Cambridge) for encouraging him to develop a theory of receptivity.
Manyofthecontributorsofthisbookarenowfacultycolleaguesindifferent parts.
It is our pleasant duty to acknowledge the contributions by Sandeep Nijhawan,
Manish Ballav, A. P. Sinha, Vivek Rana, Manoj T. Nair, K. Venkatasubbaiah,
A. K. Rao, Manojit Chattopadhyay, Z. Y. Wang, A. Dipankar, S. Sarkar, S. De,
Yogesh Bhumkar, Vijay Vedula, Manoj Rajpoot, S. Unnikrishnan, R. Bose,
N.A.Sreejith,AshishBhole,andSoumyoSengupta.Inaveryrecentdevelopment,
the authors have been looking at “exact” nonlinear theories of instability for
incompressible flows, based on disturbance mechanical energy and disturbance
enstrophy, with the former covered in the text here. However, we refrain from
providingthedetailsonthetheorybasedondisturbanceenstrophyandiscurrently
being probed for different types offlow fields and would be described elsewhere.
The authors acknowledge Aditi Sengupta and V. K. Suman for helping with
the development of this instability theory based on disturbance enstrophy [17].
Preface ix
Theauthorswouldliketoacknowledgethecompetenthelpprovidedintypingthis
materialbyMrs.BabyGaurandMrs.ShashiShukla.Specifically,Mrs.Gaurtakes
exceptional care in typing the text and preparing figures at all times. We also
acknowledge various helps provided to us by Mr. Mukesh Kumar.
Kanpur, India Tapan K. Sengupta
Jammu & Kashmir, India Swagata Bhaumik
References
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Contents
1 DNS of Wall-Bounded Turbulent Flow: An Introduction. . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Why Deterministic Study Is More Relevant than
Stochastic Approaches?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Historic Developments . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Present State of Art in the Field . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Different Transition Routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Role of Equilibrium Flow in DNS. . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 What Is Instability? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Temporal and Spatial Instability . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Some Instability Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8.1 Kelvin–Helmholtz Instability . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 DNS of Navier–Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Fluid Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Equation of Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Momentum Conservation Equation . . . . . . . . . . . . . . . . . 19
2.1.3 Energy Conservation Equation . . . . . . . . . . . . . . . . . . . . 22
2.1.4 Alternate Forms of the Energy Equation . . . . . . . . . . . . . 23
2.1.5 Vorticity Transport Equation for Incompressible
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.6 Derived Variable Formulation for 2D Incompressible
Navier–Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Spatial and Temporal Scales for Transitional
and Turbulent Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Numerical Methods for Developing DNS/LES . . . . . . . . . . . . . . . 31
2.3.1 Waves – Building Blocks of a Disturbance Field. . . . . . . 31
2.3.2 Resolution of Spatial Discretization. . . . . . . . . . . . . . . . . 34
2.3.3 High-Accuracy Compact Schemes for Evaluation of
First Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
xi