Chapter 1Basic features of turbulence
1.1 Introduction
Turbulence is very common but complicated phenomenon which can be observedin our daily life as mixing of coffee in hot water to motion of galaxies in theuniverse. Turbulence, in nature, is a complex, non-local, nonlinear, and multi-scale phenomenon which is described as last great unsolved problems of classicalphysics [2，3]. Most flows occurring in engineering applications, for example, flows inpipelines, automobile engines and flow over plane turbines, ships and aircraft areturbulent. Turbulence is also observed in human circulatory and respiratory sys?tems to atmospheric and oceanic flows. There are many other examples where wecan see turbulence, e.g., disordered behavior of the air motion exhibited by smokeof a cigarette or over a fire, water in a river or waterfall, a violent wind with abruptchanges in direction and velocity. Practically, it is important in making predictionsabout the weather, the circulation of the atmosphere and the oceans, heat transfer in nuclear reactors and drag in oil pipelines. Accuracy of astronomic observationsis suffered possibly due to small-scale turbulence in the atmospheres. It is a wellknown fact that turbulence cannot exist by itself, however; it requires a continuoussupply of energy. In Earth's atmosphere, there are at least two sources of stirring:temperature gradients that produce turbulent flows transporting heat from hot tocold regions, and wind shear that generates vortices. There are numerous otherexamples of turbulence in aeronautics, hydraulics, nuclear and chemical engineer?ing, oceanography, meteorology, astrophysics and geophysics. Thus, turbulence isa phenomena which often plays a key role in both natural and engineering fluidsystems. There are a lot of nonlinear dynamical systems, e.g., social networking,financial markets, transportation and internet which can also be viewed as analo?gous to turbulent flows. Although, turbulence is defined as a state of flow whichis disordered in space and time but to most of the researchers of turbulence, it isdifficult to confine turbulence with some good definition W. Instead of a precisemathematical definition it is preferred to describe turbulence by some commonproperties that are observed in variety of turbulent flows. Sometimes beautifulpictures are also helpful to understand turbulence.
…………
1.2 Why study turbulence?
The answer of this question lies in the widespread occurrence and extremely usefulapplications of turbulence. Turbulence is a natural phenomenon which occurs inwidespread fashion, not only on the earth but on the universe. Most of flowspossess either partially or totally turbulent states while moving. Turbulence can beobserved in respiratory and circulatory systems of living bodies, in industrial andchemical flows, in rivers, oceans and atmospheres, in geophysical and astrophysicalphenomena and in planets and galaxies.Despite the ubiquity of turbulence, the problem of turbulence is still consideredas the last unsolved problem of classical mathematical physics. For the physicist,the interesting part is how the small-scale structure of turbulence is organized,preferably isolated from any boundary effects. A complete understanding of oc?currence of turbulence and reliable prediction of its behavior in simple to complexsituations is the basic point of attraction for physists. Another interesting point inthe study of turbulence is that here universal aspects can be sought, in the sensethat they should be independent of the nature of the fluid or the geometry of theproblem. It is universality that makes turbulence an exciting research subject forphysicists and mathematicians.
……………
Chapter 2Theory of small-scale statisticsand structure functions inturbulence
2.1 Energy Cascade
The energy cascade is a statistical concept to explain the dynamics of the kineticenergy spectrum. The pictorial description of turbulence was proposed by Lewis F.Richardson in 1922.The cascade picture is based on the intuitive notion thatturbulent flows possess a hierarchical structure consisting of 'eddies' (or whorls). Inthe cascade process, as a result of successive instabilities, nonlinearity transformslarge-scale velocity circulations (eddies) into circulations (eddies) at successivelysmaller scales (sizes) until they reach such a small scale that the circulation ofthe eddies is efficiently dissipated into heat by viscosity. The inspiration behindthe Richardson's cascade picture was structure of clouds and a famous verse ofJonathan Swift about description of fleas: The number of cascade steps increases with the Reynolds number and hence in-termittency in the flow also increases. This cascade picture, with additional as?sumption about chaotic nature of cascade was used by A. N. Kolmogorov in hisfamous similarity hypotheses in 1941. Hierarchical structure of various models ofturbulence is based on the cascade picture of turbulence. Most of these modelsare not based on Navier-Stokes equations and thus termed as 'phenomenologicaTmodels. In 1949，Neumann recognized that the cascade process occurs only inFourier space and not in physical space. According to energy cascade picture en?ergy transfer, nonlinearly, from large to small scales and hence energy transfer isa one-way process. However, according to Lumley it is a tw0way process as someenergy transfers back from small to large scales.
…………
2.2 Deterministic and Stochastic Approaches
Turbulent flows occur at high Reynolds numbers and evolution of the flow fieldis extremely sensitive to small changes in initial conditions, boundary conditionsand material properties [圳 One of the basic questions arise is how to solvethis problem? In other words, which is the most appropriate approach to handleturbulence? Mainly, there are two different ways to solve this problem.According to the deterministic approach turbulence problem can be solved usinga set of partial differential equations called the Navier-Stokes equations. It is afact that by these equations turbulent flows can be described properly since theseequations contain almost all the turbulence. However, the deterministic approachcan be quite expensive in computational sense. To understand computationaltechniques in Computational Fluid Dynamics (CFD)，please refer to the one ofthe recent books.
……………
3 Experimental and Numerical Databases......... 34
3.1 Channel flow......... 34
3.2 Jet flow......... 37
3.3 Homogeneous Shear turbulence......... 39
3.4 Homogenous isotropic turbulence.........39
3.5 Chapter summary .........40
4 Longitudinal and transverse structure functions.........41
4.1 Introduction......... 42
4.2 Relation between longitudinal and transverse structure......... 45
4.3 Probability Density Functions .........49
4.4 Probability and Averages......... 50
4.5 Streamwise evolution of scaling exponents ......... 52
4.6 Isotropic ratios.........58
4.7 Chapter summary .........59
5 Hierarchical structure parameters in three dimensional turbulence......... 61
5.1 Introduction ......... 61
5.2 Hierarchical Structure model......... 62
5.3 Results and Discussion .........64
5.3.1 Probability density functions and structure functions.........65
5.3.2 Hierarchical structure parameters .........69
5.4 Chapter summary .........71
Chapter 6Detrend analysis ofReynolds-stress in fully developedturbulence
6.1 Introduction
According to the famous Rischardson-Kolmogorov's energy cascade picture, theturbulent flow is fulfilled with eddies of different spatial/temporal size at leastin the so-called inertia range the inertial range，the fluid viscosityeffect is assumed to be ignorable and a statistical equilibrium state is thus expectedThis has been recognized by using the Kolmogorov-Obukhov 5/3 law onFourier power-spectrum or equivalently the 2/3rd law for the so-called structurefunction. To show thLs multi-scale behavior of turbulent fluctuation, a 0.25 second portion oftwo velocity components from a decaying turbulent flow in a wind tunnel is shownin Figure 6.1. Visually, the longitudinal and transverse velocities are varying ondifferent time scales, indicating the multi-scale nature of the turbulent flow. Theyare positively correlated with each other on some portion and negatively on anotherportion. This could be shown more clearly by a linear trend fitting (bold blackline) within a window of second. Note that the linear trend also showspositive correlation on some portion and negative on another portion.
…………
Conclusion
Turbulence is an omnipresent but a complicated phenomenon which can be ob?served from small-scale as in a coffee cup to large-scale as the motion of galaxies.It is a complex and still an unsolved problem of classical physics due to its nonlin?ear, non-local and multi-scale nature. A large number of researchers are trying tounderstand its various aspects to put their efforts in its solution. The central aimof this thesis is to understand and add some new insights to small-scale statisticsof fully developed turbulence. The objective is achieved by studying structurefunctions and related statistics from different types of turbulent flows includingnearly isotropic decaying channel flow, jet turbulence, numerically simulated ho?mogeneous shear turbulence and homogeneous isotropic turbulence measured atvariety of Reynolds numbers.
……………
Reference (omitted)