Physics, Mathematics and Modeling

1 INTRODUCTORYLECTURES on TURBULENCEP hysics, Mathematics and ModelingJ. M. McDonoughDepartments of Mechanical Engineering and MathematicsUniversity of Kentuckyc 2004, 2007 Contents1 Fundamental Why Study Turbulence? .. Some Descriptions of Turbulence .. A Brief History of Turbulence .. General overview .. Three eras of turbulence studies .. Definitions, Mathematical Tools, Basic Concepts .. Definitions .. Mathematical tools .. Further basic concepts .. Summary .. 572 Statistical Analysis and Modeling of The Reynolds-Averaged Navier Stokes Equations .. Derivation of the RANS equations .. Time-dependent RANS equations .. Importance of vorticity and vortex stretching to turbulence .. Some general problems with RANS formulations .. Reynolds-averaged Navier Stokes Models .. The Kolmogorov Theory of Turbulence.

2 Kolmogorov s universality assumptions .. Hypotheses employed by Frisch [80] .. Principal results of the K41 theory .. Summary .. 1043 Large-Eddy Simulation and Multi-Scale Large-Eddy Simulation .. Comparison of DNS, LES and RANS methods .. The LES decomposition .. Derivation of the LES filtered equations .. Subgrid-scale models for LES .. Summary of basic LES methods .. Dynamical Systems and Multi-Scale Methods .. Some basic concepts and tools from dynamical systems theory .. The Navier Stokes equations as a dynamical system .. Multi-scale methods and alternative approaches to LES .. Summary of dynamical systems/multi-scale methods .. Summary .. 162 References163iiiCONTENTSList of da Vinci sketch of turbulent flow.. The Reynolds experiment; (a) laminar flow, (b) early transitional (but still laminar) flow,and (c) turbulence.

3 Movements in the study of turbulence, as described by Chapman and Tobak [1].. Plots of parts of Reynolds decomposition.. Turbulence energy wavenumber spectrum.. Low-pass and high-pass filtered parts of a signal.. Comparison of laminar and turbulent velocity profiles ina duct; (a) laminar, and (b) turbulent. Law of the wall.. Multiple time scales for construction of time-dependent RANS equations.. Loss of information due to averaging; (a) the complete signal, and (b) the time-averagedsignal.. Comparison of variousk models for flow over a rib of square cross section.. Drag flow over a circular cylinder, from experimental data referencedin Tritton [127].. Energy spectrum showing cut-off wavenumbers for filtered(kc) and test-filtered (k c) Energy spectrum depicting scale similarity.. Time series of a steady solution to an ODE dynamical system.

4 Phase portrait of steady attractor.. Bifurcation (transition) to convection in Rayleigh B enard problem; (a) conduction, and (b)convection.. Qualitative bifurcation diagram for Rayleigh B enardconvection.. Time series of (a) periodic and (b) subharmonic solutions to an ODE dynamical system.. Comparison of phase portraits of (a) periodic, and (b) subharmonic attractors.. Subharmonic bifurcation sequence (a) periodic, (b) subharmonic and (c) second Construction of the 1/3 Cantor set.. Geometric representation of construction of a strangeattractor.. Demonstration of scale similarity in H enon map.. (a) Sensitivity to initial conditions in the PMNS equation DDS, and (b) corresponding phaseportrait.. Example multi-scale gridding.. Example multi-scale gridding.. Grid point locations used for second-order structure function averaging in the 3-D uniform-grid case.

5 Physical model employed for synthetic-velocity LES ofswirling, buoyant plume in an Left side is sequence of experimental time series from [58]; right side presents correspondingcomputational time series from the PMNS equation DDS.. 149iiiivLIST OF PMNS plus thermal energy equation Tvs. Tbifurcation diagrams; (a) horizontal velocity,(b) vertical velocity, and (c) temperature.. Comparison of PMNS plus thermal convection DDS with DNS.. Comparison of PMNS plus thermal convection equations with high-Radata of Cioniet al.[191].. Regime map displaying possible types of time series from the PMNS equation.. Third-order structure function of PMNS time series forhomogeneous, isotropic bifurcationparameters.. Second-, fourth- and sixth-order structure functionsfrom PMNS time series for homoge-neous, isotropic bifurcation parameters.. PMNS equation 1-D energy spectrum.

6 155 Chapter 1 Fundamental ConsiderationsIn this chapter we first consider turbulence from a somewhat heuristic viewpoint, in particular discussing theimportance of turbulence as a physical phenomenon and describing the main features of turbulent flow thatare easily recognized. We follow this with an historical overview of the study of turbulence, beginning withits recognition as a distinct phenomenon by da Vinci and jumping to the works of Boussinesq and Reynoldsin the 19thCentury, continuing through important 20thCentury work of Prandtl, Taylor, Kolmogorov andmany others, and ending with discussion of an interesting paper by Chapman and Tobak [1] from thelate 20thCentury. We then provide a final section in which we begin our formal study of turbulence byintroducing a wide range of definitions and important tools and terminology needed for the remainder ofour Why Study Turbulence?

7 The understanding of turbulent behavior in flowing fluids is one of the most intriguing, frustrating and important problems in all of classical physics . It is a fact that most fluid flows are turbulent, andat the same time fluids occur, and in many cases represent the dominant physics , on all macroscopicscales throughout the known universe from the interior of biological cells, to circulatory and respiratorysystems of living creatures, to countless technological devices and household appliances of modern society,to geophysical and astrophysical phenomena including planetary interiors, oceans and atmospheres andstellar physics , and finally to galactic and even supergalactic scales. (It has recently been proposed thatturbulence during the very earliest times following the BigBang is responsible for the present form ofthe Universe.) And, despite the widespread occurrence of fluid flow, and the ubiquity of turbulence, the problem of turbulence remains to this day the last unsolved problem of classical mathematical problem of turbulence has been studied by many of the greatest physicists and engineers of the 19thand 20thCenturies, and yet we do not understand in complete detail how or why turbulence occurs, norcan we predict turbulent behavior with any degree of reliability, even in very simple (from an engineeringperspective) flow situations.

8 Thus, study of turbulence is motivated both by its inherent intellectualchallenge and by the practical utility of a thorough understanding of its Some Descriptions of TurbulenceIt appears that turbulence was already recognized as a distinct fluid behavior by at least 500 years ago(and there are even purported references to turbulence in the Old Testament). The following figure is arendition of one found in a sketch book of da Vinci, along witha remarkably modern description: ..the smallest eddies are almost numberless, and largethings are rotated only by large eddies and not by small ones,and small things are turned by small eddies and large. 12 CHAPTER 1. FUNDAMENTAL CONSIDERATIONSF igure : da Vinci sketch of turbulent phenomena were termed turbolenza by da Vinci, and hence the origin of our modern word for thistype of fluid Navier Stokes equations, which are now almost universally believed to embody the physics of allfluid flows (within the confines of the continuum hypothesis),including turbulent ones, were introduced inthe early to mid 19thCentury by Navier and Stokes.

9 Here we present these in the simple form appropriatefor analysis of incompressible flow of a fluid whose transportproperties may be assumed constant: U= 0,( )Ut+U U= P+ U+FB.( )In these equationsU= (u, v, w)Tis the velocity vector which, in general, depends on all three spatialcoordinates (x, y, z);Pis the reduced, or kinematic (divided by constant density) pressure, andFBis ageneral body-force term (also scaled by constant density).The differential operators and are thegradient and Laplace operators, respectively, in an appropriate coordinate system, with denoting thedivergence. The subscripttis shorthand notation for time differentiation, / t, and is kinematic equations are nonlinear and difficult to solve. As is well known, there are few exact solutions,and all of these have been obtained at the expense of introducing simplifying, often physically unrealistic,assumptions.

10 Thus, little progress in the understanding ofturbulence can be obtained via analyticalsolutions to these equations, and as a consequence early descriptions of turbulence were based mainly onexperimental Reynolds (circa1880) was the first to systematically investigate the transition from laminar toturbulent flow by injecting a dye streak into flow through a pipe having smooth transparent walls. Hisobservations led to identification of a single dimensionless parameter, now called theReynolds number, anddenotedRe,Re= U L ,( )that completely characterizes flow behavior in this situation. In this expression and are, respectively,the fluid properties density and dynamic a velocity scale ( , a typical value of velocity, SOME DESCRIPTIONS OF TURBULENCE3say, the average), andLis a typical length scale, , the radius of a pipe through which fluid is recall thatReexpresses the relative importance of inertial and viscous forces.