Department of Marine Science | Academics | Courses | MAR 664

TURBULENCE (MAR 664)

3 credits
Instructor: Vladimir M. Kamenkovich; office hours are by appointment

COURSE DESCRIPTION.

This is an introductory course on the theory of turbulent motions. It is directed toward graduate students in physical oceanography and meteorology, and also to those students and specialists in math, physics, and computer science, who are interested in the application of their skills in fluid mechanics and geophysics. The crux of turbulence theory is the closure problem: the number of unknowns always exceeds the number of equations. We will discuss various approaches to handle this problem. In Chapter I we will review the basic principles of fluid dynamics needed for the discussion of properties of turbulent motion. Chapters II and III will deal with general concepts of turbulence theory. Chapters IV and V will present the detailed analysis of two important special cases of turbulent motions. In Chapter VI we will review the Kolmogorov theory of the universal statistical regime of small-scale components of developed turbulence. Chapter VII will deal with peculiarities of the two-dimensional turbulence and closely related properties of the geostrophic turbulence. Finally, in Chapter VIII some of the closure models, widely used in the ocean-circulation theory, will be critically discussed.

COURSE OBJECTIVES.

1. To provide the basic concepts of turbulence theory required to understand the physical processes in the ocean and atmosphere.

2. To teach skills in applying the general results and methods of turbulence theory to the analysis of particular problems.

3. To explain relations between widely used simplified models of turbulent transports and the basic concepts of turbulence theory.

TEXTS.

Class notes with home assignments will be provided for each lecture.

The following books are recommended for supplemental reading:

Monin, A. S. and A. M. Yaglom. Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. 1, 1971, 769pp; Vol. 2, 1975, 874pp. The MIT Press (Translated from Russian).

Tennekes, H. and J. L. Lumley: A First Course in Turbulence. 1972, The MIT Press, 300pp.

1. Class participation: 15%.

2. Home assignments: 55%.

3. Final exam: 30%

The final exam consists mainly of a term project. The term project is submitted in written form and is defended orally. It must contain a critical review of the published material with some elements of independent research.

PREREQUISITES.

Basic courses on physics (PHY 351) and calculus (MAT 385) or permission of the instructor.

DETAILED COURSE OUTLINE.

Chapter I. FUNDAMENTALS OF FLUID DYNAMICS (10 Lectures)

We will start with the discussion of general principles of fluid dynamics. Particular emphasis will be given to the analysis of general laws governing the thermal conduction, salt diffusion, and friction in the fluid. For what follows it is of great value to disclose the physical basis of the relations between the corresponding fluxes and gradients of basic characteristics. The vorticity dynamics will be considered in detail. Their properties are important for understanding the salient features of turbulent motion.

I.1. Deformation and rotation of a fluid particle. Vorticity.

I.2. Stokes theorem.

I.3. Equation of mass conservation. The concept of flux form of the basic equation. The evolution form of the equation.

I.4. Equation of salt diffusion.

I.5. Momentum equations (Newton's law of motion).

I.6. Angular momentum equations.

I.7. Thermodynamics of sea water.

I.8. Equation of energy conservation. The First Principle of thermodynamics.

I.9. Energy transformations. Equations for the mechanical and internal energies. Adiabatic motion.

I.10. Entropy equation. The Second Principle of thermodynamics.

I.11. Thermal conduction, diffusion, and friction in the fluid. The relation between fluxes and gradients of basic characteristics

I.12. Vorticity dynamics. Kelvin's theorem. Vorticity equation.

I.13. Boussinesq approximations.

Chapter II. THE NATURE OF TURBULENCE (4 Lectures)

First, we will list those properties of motion that are characteristic just for the turbulent motion. Then we will discuss the Reynolds decomposition of flow characteristics into the mean and fluctuating components and different methods of averaging. The turbulent transports of momentum and heat for two simple types of motion will be estimated. The detailed comparison of the turbulent and molecular transports of momentum (friction) will be presented. The general statistical formulation of the problems of turbulence theory will be given and some important relevant concepts of the theory of random functions will be discussed.

II.1. Properties of turbulent motions. The origin of turbulence.

II.2. Reynolds decomposition of flow characteristics. Averaging.

II.3. Some estimates of the intensity of turbulent mixing. Coefficient of turbulent mixing.

II.4. Molecular and turbulent transports of momentum.

II.5. Random fields and ensemble (probability) means. Statistical moments.

II.6. Statistical formulation of the problems of turbulence theory.

II.7. Stationary random functions and homogeneous fields. Ergodicity.

II.8. Spectral representations of random stationary functions and homogeneous fields.

Chapter III. BASIC EQUATIONS OF TURBULENCE THEORY (4 Lectures)

First, we will discuss the averaging of the basic fluid-dynamical equations and the ensuing closure problem. Then we will analyze the factors that influence the intensity of temperature and velocity fluctuations. The important nondimensional parameter - the Richardson number - will be introduced based on the energy consideration. The equations for the Reynolds stresses will be derived and the tendency toward isotropization caused by the pressure fluctuations will be analyzed. The physical mechanisms responsible for the enstrophy generation in turbulent flows will be thoroughly investigated.

III.1. Averaging of the basic fluid-dynamical equations. The closure problem.

III.2. Equation for the mean square of temperature fluctuations.

III.3. Equation for the kinetic energy of turbulence. The Richardson number. Taylor's microscale.

III.4. Equations for the Reynolds stresses. Rotta's hypothesis.

III.5. Equation for the enstrophy of turbulence.

Chapter IV. TURBULENT BOUNDARY LAYER (3 Lectures)

We will start with the detailed analysis of the turbulent flow within a constant-stress layer near a rigid wall (without buoyancy effects). Then the buoyancy forces will be included into consideration and the Monin-Obukhov scale will be introduced. In conclusion, we will consider the dynamics of the planetary boundary layer. In this relation, we will discuss an approach to the closure of the system of governing equations, based on the incorporation of the equation for turbulence kinetic energy.

IV.1. Turbulent flow near a rigid wall. Logarithmic layer.

IV.2. Thermally stratified turbulent boundary layer. The Monin-Obukhov scale.

IV.3. Planetary boundary layer. Parameterization of the turbulent transport of momentum.

Chapter V. ISOTROPIC TURBULENCE (3 Lectures)

The concept of homogeneous and isotropic turbulence introduced by G. I. Taylor plays a significant role in turbulence theory. We will present the detailed analysis of properties of the homogeneous and isotropic turbulence based on the von Karman-Howarth equation. Special attention will be given to the kinetic energy and enstrophy equations and to the properties of the spectral transfer of energy.

V.1. Isotropic turbulence and its experimental realization.

V.2. Correlation tensor. The von Karman-Howarth equation.

V.3. Some consequencies of the von Karman-Howarth equation.

V.4. Spectral form of the von Karman-Howarth equation.

V.5. Kolmogorov's hypothesis on self-similarity of small scales.

Chapter VI. STATISTICAL REGIME OF THE SMALL-SCALE COMPONENTS OF DEVELOPED TURBULENCE (2 Lectures)

We will analyze the turbulent flows with sufficiently large Reynolds numbers. The Kolmogorov theory of the universal statistical regime of the small-scale components of such flows will be outlined. The important concepts of the structure function and the spectrum of the locally isotropic motion are considered. We will conclude with the derivation and discussion of the famous "two-thirds" and "five-thirds laws".

VI.1. The structure of developed turbulence.

VI.2. Definition of locally isotropic turbulence.

VI.3. Kolmogorov's similarity hypotheses.

VI.4. Statistical characteristics of locally isotropic turbulence.

VI.5. Local structure of the velocity fluctuations.

Chapter VII. TWO-DIMENSIONAL TURBULENCE (2 Lectures)

Due to the energy and enstrophy conservation laws, the properties of the two-dimensional turbulence differ substantially from those of the three-dimensonal turbulence. The peculiarities of the energy and enstrophy fluxes and the corresponding spectra will be discussed in detail. Based on the consideration of the two-dimensional turbulence, we will thoroughly analyze the features of the geostrophic turbulence.

VII.1. Inviscid isotropic case.

VII.2. Spectra of 2D-turbulence.

VII.3. Energy transfer in 2D- and 3D-turbulence.

VII.4. Geostrophic turbulence. Macroturbulence.

Chapter VIII. SOME APPROACHES TO THE CLOSURE PROBLEM IN THE OCEAN-CIRCULATION THEORY (3 Lectures)

First, we will formulate the general constraints that must be satisfied by any closure model. Then the closure schemes of two widely used numerical models of the ocean circulation will be critically discussed (NLOM and POM). We will conclude with a brief consideration of the Mellor-Yamada closure models of different levels.

VIII.1. General constraints.

VIII.2. Naval Research Laboratory Layered Ocean Model (NLOM).

VIII.3. Princeton Ocean Model (POM). Coefficients of mixing.

VIII.4. Second order closure. Mellor-Yamada closure models.

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