Table Of ContentWolfgang Kollmann
Navier–Stokes
Turbulence
Theory and Analysis
–
Navier Stokes Turbulence
Wolfgang Kollmann
–
Navier Stokes Turbulence
Theory and Analysis
123
WolfgangKollmann
Department ofMechanical Engineering
University of California Davis
Davis, CA,USA
ISBN978-3-030-31868-0 ISBN978-3-030-31869-7 (eBook)
https://doi.org/10.1007/978-3-030-31869-7
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Preface
Thesubjectofthebookisthetheoreticaltreatmentoffullydevelopedturbulencein
fluids governed by the Navier–Stokes equations. The investigation begins with a
collection of properties of turbulent flows of Newtonian fluids observed in exper-
iments and direct numerical simulations. The main purpose of this assembly is to
condense them to a working definition of turbulence. Then the variables for the
complete description of turbulence based on Navier–Stokes dynamics are estab-
lished and the equations governing them are derived within the framework of
classical mechanics.
The phenomenon of turbulence is analysed with the tools of probability theory.
The fundamental difficulty encountered in this approach is the fact that countably
infinitesystemsofequationsorfunctionsofinfinitelymanyvariablesappear,which
cannot be truncated without arbitrary measures to generate solvable systems. It is,
however,possibletoderiveasingle,linearequationforthecharacteristicfunctional
containing all relevant information on a turbulent flow. This equation, derived by
E.Hopfmorethan50yearsago,iscentraltothetreatmentofturbulentflowsinthis
book.Itcontainsfunctionaloperationsthatrequirecarefulanddetailedconsideration
for their proper use. The basic operations of differentiation and integration of
functionalsarereviewedandexamplesrelevanttoturbulentflowsareconstructedto
illustratetheirapplication.Theequationforthecharacteristicfunctionalinthespatial
description contains the effects of convection as second variational derivatives,
whose trace is a generalized Laplacian. There have been several variants of the
infinite-dimensional Laplacian developed for different purposes, two of those defi-
nitions are considered in detail. These are the generalized and the Lévy-Laplacian.
Therelationsofconvection,expressedintermsofthecharacteristicfunctional,tothe
generalized Hessian and thegeneralized Laplacian operatorsare explored.
The Hopffde for turbulent flows of an incompressible fluid in statistical steady
stateisanalysedwiththeaidofasolenoidalONSbasisforacompactflowdomain
with a combination of homogeneous and periodic boundary conditions. The
argumentsofthecharacteristicfunctionalarerepresentedinthesolenoidal basisas
divergence-free vector fields thus allowing elimination of the pressure gradient in
theHopfequation.Basisprojectionoftheargumentfieldsandtheprojectionofthe
v
vi Preface
Hopffdeonto finite-dimensionalsubspacesiscentral tothenumericaltreatmentof
turbulent flow through pipes. It is shown to produce linear, second-order standard
pdes. The example of pipe flow with periodic entrance and exit conditions is the
vehicle to develop and illustrate the general theory.
Two aspects of turbulence theory require the geometric and limit properties
of the n-dimensional ball in Euclidian space as the dimension n goes to infinity,
first, the definition and evaluation of the standard and the Lévy-Laplacian;
second, the formulation of integrals in infinite-dimensional spaces.
The properties of the n-dimensional ball in Euclidian space are summarized in
Sect. 23.11 of Appendix A.
The properties of turbulence measures and the associated characteristic func-
tional are discussed at an elementary level in Chap. 6. Methods of construction of
measuresininfinite-dimensionalspacesarepresentedandillustratedwithexamples.
The Hopf and Lewis–Kraichnan equations (Vishik et al. [1, 2], Lewis and
Kraichnan [3], Dopazo and O’Brien [4]) governing the evolution of characteristic
functionalsarederivedinseveralversions.Furthermore,variantsoftheseequations
are established using various local and global mappings and a general mapping
equation is derived.
Theindeterminatepdes(partialdifferentialequations)forfinite-dimensionalPdfs
play an important role in the practical treatment of turbulence, they are deduced
using the (coarse-grained) Dirac pseudo-function leading to the well-known
Lundgren–Monin–Novikov hierarchy. The properties of the terms representing
convection, frictional effects and external volume forces are established and
examples are constructed to illustrate their properties. Furthermore, examples for
the closure of these pdes are presented and the shortcomings of various arbitrary
assumptions necessary to construct a solvable system of equations are discussed.
Homogeneous turbulence is discussed in the light of the hypotheses of
Kolmogorov and Onsager and recent experimental results leading to modifications
ofthetheoryduetointernalintermittency.Thenotionofstructuresinturbulentflow
fields and their classification is analysed in the spatial and material descriptions.
Several examples are discussed in detail.
Wall bounded flows are of prime importance for theoretical and a variety of
practicalreasons.Thus,particularexamples(pipeflowandplanechannelflow)are
chosen to illustrate theoretical difficulties and, briefly in Appendices C and D,
simple engineering approaches for dealing with this phenomenon are presented.
Theemphasisisplacedonfullydevelopedturbulenceforincompressiblefluids,
butsomeaspectsofcompressiblefluidsarealsoincludedinSects.2.1,2.1.2,2.5.3,
while stability and transitional phenomena are not covered, since they deserve a
volume on their own and there is an enormous amount of literature on these
subjects available [5–9], Chap. 9 and references therein.
Davis, CA, USA Wolfgang Kollmann
Preface vii
References
1. Vishik, M.J.: Analytic solutions of Hopf’s equation corresponding to quasilinear parabolic
equationsortotheNavier-Stokessystem.Sel.Math.Sov.5,45–75(1986)
2. Vishik,M.J.,Fursikov,A.V.:MathematicalProblemsofStatisticalHydromechanics.Kluwer
AcademicPubl.,Dordrecht(1988)
3. Lewis,R.M.,Kraichnan,R.H.:Aspace-timefunctionalformalismforturbulence.Comm.Pure
Appl.Math.XV,397–411(1962)
4. Dopazo,C.,O’Brien,E.E.:Functionalformulationofnonisothermalturbulentreactiveflows.
Phys.Fluids17,1968–1975(1974)
5. Schlichting,H.:BoundaryLayerTheory.McGraw-Hill(1987)
6. Ruelle,D.,TakensF.:Onthenatureofturbulence.Comm.Math.Phys.20,167–192(1971)
7. Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical
systems.Comm.Math.Phys.74,189–197(1980)
8. Andereck, C.D., Liu, S.S., Swinney, H.L.: Flow regimes between independently rotating
cylinders.JFM164,155–183(1986)
9. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer Verlag
(2001)
Acknowledgements
It is a pleasure to acknowledge many fruitful discussions with J. J. Chattot (MAE
Department, University of California, Davis), M. S. Chong (MAE Department,
University of Melbourne, Australia), J. Janicka (EKT, TU-Darmstadt, Germany),
M. Oberlack (Department of Mechanical Engineering, TU-Darmstadt, Germany),
B. A. Younis (Civil and Environmental Engineering, UCD) and M. D. White
(Dayton, Ohio). I am particularly indebted to Florian Ries (Department of
Mechanical Engineering,TU-Darmstadt)forproviding DNSdata for thepipe flow
and to Thomas O. Kollmann (Davis) for help with graphics applications.
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Navier–Stokes Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Spatial/Eulerian Description. . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 Symmetries of the Euler and Navier–Stokes pdes . . . 22
2.2 Fundamental Properties of the Solutions for a Single
Incompressible Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Rotation and Vorticity in the Spatial Description . . . . . . . . . . 33
2.3.1 Beltrami and Trkalian Vector Fields . . . . . . . . . . . . 35
2.3.2 Vorticity pde for Compressible Fluids . . . . . . . . . . . 35
2.4 Lamb Vector Dynamics for Incompressible Fluids . . . . . . . . . 37
2.5 Material/Lagrangean Description . . . . . . . . . . . . . . . . . . . . . . 40
2.5.1 Piola Transform … . . . . . . . . . . . . . . . . . . . . . . . . 41
ab
2.5.2 Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.3 Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Rotation and Vorticity in the Material Description . . . . . . . . . 44
2.6.1 Divergence of the Material Vorticity . . . . . . . . . . . . 45
2.6.2 Material Vorticity in Plane Flows . . . . . . . . . . . . . . 46
2.6.3 Frozen Vector Fields. . . . . . . . . . . . . . . . . . . . . . . . 47
2.7 Velocity Gradient Tensor in the Spatial Description . . . . . . . . 49
2.8 Problems for this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Basic Properties of Turbulent Flows. . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Working Definition of Turbulence . . . . . . . . . . . . . . . . . . . . . 63
3.2 Asymptotic Properties of Turbulent Flows . . . . . . . . . . . . . . . 66
xi
xii Contents
3.3 Number of Degrees of Freedom of the Turbulence
Attractor for Maintained Turbulence. . . . . . . . . . . . . . . . . . . . 67
3.4 Problems for this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Flow Domains and Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Homogeneous Turbulence Domain Dht. . . . . . . . . . . . . . . . . . 74
4.2 Periodic Pipe Flow Domain D. . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Schauder Basis for Scalar Fields . . . . . . . . . . . . . . . 76
4.2.2 Schauder Basis for Vector Fields with Non-zero
and Zero Dilatation. . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Open and Noncompact Domains . . . . . . . . . . . . . . . . . . . . . . 80
4.3.1 Semi-infinite Pipe: z ¼0 and z ¼1. . . . . . . . . . . 80
0 1
4.3.2 Boundary Layer Domain Dbl. . . . . . . . . . . . . . . . . . 82
4.3.3 Free Shear Layer Domain Dfs . . . . . . . . . . . . . . . . . 85
4.4 Problems for this Chapter: Flow Domains and Bases . . . . . . . 88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Phase and Test Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 Solution Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Phase Space for the Turbulence Measure: Incompressible
Fluids and Homogeneous Boundary Conditions . . . . . . . . . . . 94
5.3 Phase Space: Noncompact Domains. . . . . . . . . . . . . . . . . . . . 96
5.4 Argument/Test Function Space for Characteristic
Functionals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Example for a Nuclear Space. . . . . . . . . . . . . . . . . . . . . . . . . 98
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Probability Measure and Characteristic Functional . . . . . . . . . . . . 101
6.1 Elementary Properties of the Turbulence Measure. . . . . . . . . . 102
6.2 Turbulence Measure for Incompressible Fluids . . . . . . . . . . . . 104
6.3 Construction of Measures as Limits of Cylinder Measures. . . . 105
6.3.1 Cylinder Measures . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.2 Kolmogorov Extension Theorem . . . . . . . . . . . . . . . 106
6.3.3 Examples of Measures in Function Spaces. . . . . . . . 108
6.3.4 Wiener Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4 Problems for this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7 Functional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.1 Elliptic fdes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Parabolic fdes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3 Problems for this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
