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Scientific Computation EditorialBoard J.-J.Chattot,Davis,CA,USA P.Colella,Berkeley,CA,USA WeinanE,Princeton,NJ,USA R.Glowinski,Houston,TX,USA M.Holt,Berkeley,CA,USA Y.Hussaini,Tallahassee,FL,USA P.Joly,LeChesnay,France H.B.Keller,Pasadena,CA,USA D.I.Meiron,Pasadena,CA,USA O.Pironneau,Paris,France A.Quarteroni,Lausanne,Switzerland J.Rappaz,Lausanne,Switzerland R.Rosner,Chicago,IL,USA. J.H.Seinfeld,Pasadena,CA,USA A.Szepessy,Stockholm,Sweden M.F.Wheeler,Austin,TX,USA L. C. Berselli T. Iliescu W. J. Layton Mathematics of Large Eddy Simulation of Turbulent Flows With32Figures 123 Dr.LuigiC.Berselli Dr.WilliamJ.Layton UniversityofPisa UniversityofPittsburgh DepartmentofAppliedMathematics DepartmentofMathematics “U.Dini” ThackerayHall301 ViaBonanno25/b Pittsburgh,PA15260,USA I-56126Pisa,Italy e-mail:[email protected] e-mail:[email protected] Dr.TraianIliescu VirginiaPolytechnicInstitute andStateUniversity DepartmentofMathematics 456McBrydeHall Blacksburg,VA24061,USA e-mail:[email protected] LibraryofCongressControlNumber:2005930495 ISSN 1434-8322 ISBN-10 3-540-26316-0 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-26316-6 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductionon microfilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationorpartsthereofispermitted onlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermission forusemustalwaysbeobtainedfromSpringer.ViolationsareliableforprosecutionundertheGermanCopyright Law. SpringerisapartofSpringerScience+BusinessMedia springeronline.com ©Springer-VerlagBerlinHeidelberg2006 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenin theabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsand thereforefreeforgeneraluse. Typesetting:DataconversionbyLE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig,Germany Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper 55/3141/YL 543210 To Lucia, Raffaella, and Annette Preface Turbulence is ubiquitous in nature and central to many applications important to our life. (It is also a ridiculously fascinating phenomenon.) Obtaining an accurate prediction of turbulent flow is a central difficulty in such diverse problems as global change estimation, improving the energy efficiency of en- gines,controllingdispersalofcontaminantsanddesigningbiomedicaldevices. Itisabsolutelyfundamentaltounderstandingphysicalprocessesofgeophysics, combustion, forces of fluids upon elastic bodies, drag, lift and mixing. Deci- sionsthataffectourlifemustbemadedailybasedonpredictionsofturbulent flows. Direct numerical simulation of turbulent flows is not feasible for the fore- seeablefuture inmany ofthese applications.Evenfor thoseflows forwhichit is currently feasible,it is filled with uncertainties due to the sensitivity of the flow to factors such as incomplete initial conditions, body forces, and surface roughness.Itisalsoexpensiveandtimeconsuming–fartootimeconsumingto use as a designtool.Storing,manipulating andpost-processingthe mountain of uncertain data that results from a DNS to extract that which is needed from the flow is also expensive, time consuming, and uncertain. The most promising and successful methodology for doing these simula- tionsofthat which matters inturbulentflowsislargeeddysimulationorLES. LES seeks to calculate the large, energetic structures (the large eddies) in a turbulent flow. The aim of LES is to do this with complexity-independent of the Reynolds number and dependent only on the resolution sought. The approach of LES, developed over the last 35 years, is to filter the Navier– Stokes equations, insert a closure approximation (yielding an LES model), supply boundary conditions (called a Near Wall Model in LES), discretize appropriatelyandperformasimulation.ThefirstthreekeychallengesofLES are thus: Do the solutions of the chosen model accurately reflect true flow averages? Do the numerical solutions generated by the chosen discretization, reflect solutions of the model? And, With the chosen model and method, how is simulation to be performed in a time and cost effective manner? Although allthree questions are consideredherein, we havefocused mostly on the first, VIII Preface i.e.themathematicaldevelopmentoftheLESmodelsthemselves.Thesecond and third questions concerning numerical analysis and computational simulation of LES models are essential. However, the numerical analysis of LES should not begin by assuming a model is a correct mathematical realization of the intended physical phenomenon (in other words, that the model is well posed). To do so would be to build on a foundation of optimism. Numerical analysis of LES models with sound mathematical foundations is an exciting challenge for the next stage of the LES adventure. One important approach to unlocking the mysteries of turbulence is by computational studies of key, building block turbulent flows (as proposed by vonNeumann).ThegreatsuccessofLESineconomicalandaccuratedescrip- tionsofmanybuildingblockturbulentflowshassparkeditsexplosivegrowth. Its development into a predictive tool, useful for control and design in com- plexgeometries,isclearlythenextstep,andpossiblywithinreachinthenear future. This development will require much more experience with practical LESmethods.Itwillalsorequirefundamentalmathematicalcontributionsto understanding “How”, “Why”, and “When” an approach to LES can work and “What” is the expected accuracy of the combination of filter, model, discretization and solver. The extension of LES from application to fully developed turbulence to includetransitionandwalleffectsandthentothedelicateproblemsofcontrol anddesignisclearlythenextstepinthedevelopmentoflargeeddysimulation. Progress is already being made by careful experimentation. Even as “[The universe] is written in mathematical language” (Galileo), the Navier–Stokes equations are the language of fluid dynamics. Enhancing the universality of LES requiresmaking a direct connectionbetween LESmodels and the (often mathematicallyformidable)Navier–Stokesequations.Onethemeofthisbook istheconnectionbetweenLESmodelsandtheNavier–Stokesequationsrather than the phenomenology of turbulence. Mathematical development will complement numerical experimentation and make LES more general, universal, robust and predictive. Wehavewrittenthisbookinthehopeitwillbeuseful forLESpractition- ersinterestedinunderstandinghowmathematicaldevelopmentofLESmodels can illuminate models and increase their usefulness, for applied mathemati- cians interestedinthe areaandespecially for PhDstudents incomputational mathematics trying to make their first contribution. One of the themes we emphasize is that mathematical understanding, physical insight and computational experience are the three foundations of LES! Throughout, we try to present the first steps of a theory as simply as possible, consistent with cor- rectness and relevance, and no simpler. We have tried, in this balancing act, to find the right level of detail, accuracy and mathematical rigor. This book collects some of the fundamental ideas and results scattered throughoutthe LESliteratureandembedstheminahomogeneousandrigor- ous mathematical framework. We also try to isolate and focus on the mathematical principles shared by apparently distinct methodologies in LES and Preface IX show their essential role in robust and universal modeling. In part I we re- view basic facets of on the Navier–Stokes equations; in parts II and III we highlight some promising models for LES, giving details of the mathematical foundation, derivation and analysis. In part IV we present some of the diffi- cult challenges introduced by solid boundaries;part V presents a syllabus for numerical validation and testing in LES. We are all too aware of the tremendous breadth, depth and scope of the area of LES and of the great limitations of our own experience and understanding. Some of these gaps are filled in other excellent books on LES. In particular, we have learned a lot ourselves from the books of Geurts [131], John [175], Pope [258], and Sagaut [267]. We have tried to complement the treatment of LES in these excellent books by developing mathematical tools, methods,andresultsforLES.Thus,manyofthesametopicsareoftentreated hereinbutwiththemagnifyingglassofmathematicalanalysis.Thistreatment yields new perspectives, ideas, language and illuminates many open research problems. We offer this book in the hope that it will be useful to those who will help developthe field of LESand fill in many of the gaps wehaveleft behind herein. It is a pleasure to acknowledge the help of many people in writing this book. We thank Pierre Sagaut for giving us the initial impulse in the project andformanydetailedandhelpfulcommentsalongtheway.Weoweourfriend and colleague Paolo Galdi a lot as well for many exciting and illuminating conversations on fluid flow phenomena. Our first meeting came through one such interactionwith Paolo.We also thank VolkerJohn, who throughoutour LES adventure has been part of our day to day “battles”. OurunderstandingofLEShasadvancedthroughworkingwithfriendsand collaborators Mihai Anitescu, Jeff Borggaard, Adrian Dunca, Songul Kaya, Roger Lewandowski, and Niyazi Sahin. The preparation of this manuscript has benefited from the financial sup- port of the National Science Foundation, the Air Force office of Scientific Research, and Ministero dell’Istruzione, dell’Universita` e della Ricerca. Pisa, Italy Luigi C. Berselli Blacksburg, USA Traian Iliescu Pittsburgh, USA William J. Layton April, 2005 Index of Acronyms AD, Approximate deconvolution, 111 ADBC, Approximate deconvolution boundary conditions, 258 BCE, Boundary commutation error, 245 CFD, Computational fluid dynamics, 118 CTM, Conventional turbulence model, 8 DNS, Direct numerical simulation, 5 EV, Eddy viscosity, 20 FFT, Fast Fourier transform, 321 GL, Gaussian–Laplacian,112 LES, Large Eddy Simulation, 3 NSE, Navier–Stokes equations, 3 NWM, Near wall model, 253 SFNSE, Space filtered Navier–Stokes equations, 16 SFS, Subfilter-scale stresses, 135 SLM, Smagorinsky–Ladyˇzhenskayamodel, 81 SNSE, Stochastic Navier–Stokes equations, 65 VMM, Variational multiscale method, 28 Contents Part I Introduction 1 Introduction............................................... 3 1.1 Characteristics of Turbulence ............................. 6 1.2 What are Useful Averages? ............................... 8 1.3 Conventional Turbulence Models .......................... 14 1.4 Large Eddy Simulation................................... 16 1.5 Problems with Boundaries................................ 17 1.6 The Interior Closure Problem in LES ...................... 18 1.7 Eddy Viscosity Closure Models in LES ..................... 20 1.8 Closure Models Based on Systematic Approximation......... 22 1.9 Mixed Models........................................... 25 1.10 Numerical Validation and Testing in LES................... 26 2 The Navier–Stokes Equations.............................. 29 2.1 An Introduction to the NSE .............................. 29 2.2 Derivation of the NSE ................................... 32 2.3 Boundary Conditions .................................... 36 2.4 A Few Results on the Mathematics of the NSE.............. 37 2.4.1 Notation and Function Spaces ..................... 38 2.4.2 Weak Solutions in the Sense of Leray–Hopf .......... 42 2.4.3 The Energy Balance .............................. 43 2.4.4 Existence of Weak Solutions ....................... 47 2.4.5 More Regular Solutions ........................... 54 2.5 Some Remarks on the Euler Equations ..................... 62 2.6 The Stochastic Navier–Stokes Equations.................... 65 2.7 Conclusions............................................. 68 XIV Contents Part II Eddy Viscosity Models 3 Introduction to Eddy Viscosity Models .................... 71 3.1 Introduction ............................................ 71 3.2 Eddy Viscosity Models ................................... 72 3.3 Variations on the Smagorinsky Model ...................... 77 3.3.1 Van Driest Damping.............................. 78 3.3.2 Alternate Scalings ................................ 78 3.3.3 Models Acting Only on the Smallest Resolved Scales.. 80 3.3.4 Germano’s Dynamic Model ........................ 80 3.4 Mathematical Properties of the Smagorinsky Model.......... 81 3.4.1 Further Properties of Monotone Operators........... 93 3.5 Backscatter and the Eddy Viscosity Models.................102 3.6 Conclusions.............................................103 4 Improved Eddy Viscosity Models ..........................105 4.1 Introduction ............................................105 4.2 The Gaussian–LaplacianModel (GL) ......................111 4.2.1 Mathematical Properties ..........................112 4.3 k−ε Modeling..........................................117 4.3.1 Selective Models .................................118 4.4 Conclusions.............................................121 5 Uncertainties in Eddy Viscosity Models and Improved Estimates of Turbulent Flow Functionals .............................123 5.1 Introduction ............................................123 5.2 The Sensitivity Equations of Eddy Viscosity Models .........124 5.2.1 Calculating f = ∂ f..............................126 δ ∂δ 5.2.2 Boundary Conditions for the Sensitivities............127 5.3 Improving Estimates of Functionals of Turbulent Quantities ..127 5.4 Conclusions: Are u and p Enough? ........................130 Part III Advanced Models 6 Basic Criteria for Subfilter-scale Modeling.................135 6.1 Modeling the Subfilter-scale Stresses .......................135 6.2 Requirements for a Satisfactory Closure Model ..............136