4A8: Environmental Fluid Mechanics
Mixing and Reactions in Turbulent Flows
3. Statistical description of turbulent mixing
In this Chapter, we will derive the governing equation for a reacting scalar in a turbulent
flow and we will demonstrate why the turb
Approach to Chaos 0. Introduction There are two major reasons for studying non-linear systems. The first and most basic is that the equations of motion of almost all real systems are non-linear. The second reason is that even a relatively simple system wh
Chapter 5
Free thin shear ows
In this Chapter, we will discuss a large class of ows called thin shear ows. We will focus
on ows far from a solid surface (the wall boundary layer is discussed in Chapter 6) and
we will aim to understand the structure of jet
Chapter 3
Governing equations
3.1
Instantaneous equations (incompressible uid)
Conservation of mass
Navier-Stokes
ui
=0
xi
(3.1)
1 p
1 ik
ui (uk ui )
=
+ gi +
+
t
xk
xi
xk
(3.2)
with the shear stress given by:
ij = 2sij =
ui uj
+
xj
xi
(3.3)
with sij
Cambridge University Engineering Department
4A12: Turbulence
Lecture Notes
Dr. E. Mastorakos
Hopkinson Lab
E-mail: em257@eng.cam.ac.uk
http:/www.eng.cam.ac.uk/ em257
1
Chapter 1
Introduction
1.1
Aims
The main aims of this course are:
1. To introduce some
Turbulent Flow and Transport
6
Introduction to Turbulent Boundary Layers
6.1
The nature of flow in turbulent boundary layers. Inner and outer regions, eddy
diffusivity distributions, intermittency, etc.
6.2
Integral form of the mean flow boundary layer e
Tony Burdens Lecture Notes on Turbulence, Spring 2006
Wall-bounded shear ows
version 1: channel ow
Fully developed channel ow
See the example in the section of the lecture notes which present Reynolds equation
and the Reynolds stress.
For fully developed,
Lecture Notes. Waves in Random Media
Guillaume Bal
1
January 9, 2006
1
Department of Applied Physics and Applied Mathematics, Columbia University, New York NY,
10027; gb2030@columbia.edu
Contents
1 Wave equations and First-order hyperbolic systems
1.1 Int
FUNDAMENTAL AND CONCEPTUAL
ASPECTS OF TURBULENT FLOWS
Arkady Tsinober
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows
Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectu
FUNDAMENTAL AND CONCEPTUAL
ASPECTS OF TURBULENT FLOWS
Arkady Tsinober
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows
Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectu
FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectures series as a
FUNDAMENTAL AND CONCEPTUAL
ASPECTS OF TURBULENT FLOWS
Arkady Tsinober
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows
Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectu
FUNDAMENTAL AND CONCEPTUAL
ASPECTS OF TURBULENT FLOWS
Arkady Tsinober
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows
Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectu
FUNDAMENTAL AND CONCEPTUAL
ASPECTS OF TURBULENT FLOWS
Arkady Tsinober
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows
Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectu
FUNDAMENTAL AND CONCEPTUAL
ASPECTS OF TURBULENT FLOWS
Arkady Tsinober
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows
Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectu
FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectures series as a
FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS
Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectures series as a
Chapter 4 Turbulent ows
4.1 Transition to turbulence
In the case of free ows transition to turbulence occurs much earlier than in conned ows. In terms of Reynolds number, it is a matter of several hundred for the unbounded ows around objects, and a matter
Chapter 3 Laminar ows
3.1 Assumptions
Flow equations discussed in Chapter 2 provide analytical solutions only in some special cases. In this chapter we shall consider the equations for incompressible ow: (2.4), (2.22) and (2.74), assuming that all the coe
42
CHAPTER 2. FUNDAMENTAL LAWS
where is the density of the uid and vn is the face-normal velocity across the face of area A of a control volume (see Problem 2.7.7).
2.6 The Law of Similarity
The law of similarity [7, 8] enables in some situations to use a
Chapter 14
Turbulence
Version 0214.2, 5 February 2003
Please send comments, suggestions, and errata via email to kip@tapir.caltech.edu and also
to rdb@caltech.edu, or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125
14.1
Overview
In Chap. 12, we
Chapter 12
Foundations of Fluid Dynamics
Version 0212.2 28 Jan 03
Please send comments, suggestions, and errata via email to kip@tapir.caltech.edu and rdb@caltech.edu,
or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125
12.1
Overview
Having studi
Turbulent Flow and Transport
9
Dispersion in Pipe and Channel flow
9.1
Dispersion in laminar pipe flow. Purely diffusive dispersion, purely
convective dispersion, and Taylor (or TaylorAris) dispersion. Scaling laws that define
the conditions under which t
Turbulent Flow and Transport
8
Introduction to Turbulence Models
8.1
Approaches to closure. Eddy diffusivity defined in terms of local turbulence
length scale.
8.2
Equations for (i) the kinetic energy of the mean motion and for (ii) the mean kinetic ene
Turbulent Flow and Transport
7
7.1 7.2 7.3 S.
Buoyant Plumes, Thermals, etc.
Jets, plumes, forced plumes, thermals, etc. Stable and unstable atmospheres. The Boussinesq approximation and the conditions for its applicability. Integral equations for a stea
Introduction to Turbulence
by
Hkan Gustavsson
Division of Fluid Mechanics
Lule University of Technology
Foreword
The intention with these pages is to present the student with the basic theoretical concepts of
turbulence and derive exact relations from the
Turbulent Mixing in the Microscale
W.Dzwinel1, W.Alda1, M.Pogoda1 and D.A.Yuen2
1
University of Mining and Metallurgy, Institute of Computer Science, 30-059 Krakw, Poland
2
Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, USA
Abstr