Analysis of Turbulent Flows with Computer Programs

Tuncer Cebeci

• Publisher: Elsevier Science
• Year: 2004
• eBook ISBN: 9780080527185
• Topic: Technology & Engineering
• Subtopic: Aeronautic & Astronautic Engineering
• Index: Technology & Engineering

Modelling and Computation of Turbulent Flows has been written by one of the most prolific authors in the field of CFD. Professor of aerodynamics at SUPAERO and director of DMAE at ONERA, the author calls on both his academic and industrial experience when presenting this work. The field of CFD is strongly represented by the following corporate companies; Boeing; Airbus; Thales; United Technologies and General Electric, government bodies and academic institutions also have a strong interest in this exciting field. Each chapter has also been specifically constructed to constitute as an advanced textbook for PhD candidates working in the field of CFD, making this book essential reading for researchers, practitioners in industry and MSc and MEng students.* A broad overview of the development and application of Computational Fluid Dynamics (CFD), with real applications to industry* A Free CD-Rom which contains computer program's suitable for solving non-linear equations which arise in modeling turbulent flows* Professor Cebeci has published over 200 technical papers and 14 books, a world authority in the field of CFD

1

Introduction

1.1 Introductory Remarks

Turbulence in viscous flows is described by the Navier–Stokes equations, perfected by Stokes in 1845, and now soluble by Direct Numerical Simulation (DNS). However, computing capacity restricts solutions to simple boundary conditions and moderate Reynolds numbers and calculations for complex geometries are very costly. Thus, there is need for simplified, and therefore approximate, calculations for most engineering problems. It is instructive to go back some eighty years to remarks made by Prandtl [1] who began an important lecture as follows:

What I am about to say on the phenomena of turbulent flows is still far from conclusive. It concerns, rather, the first steps in a new path which I hope will be followed by many others.

The researches on the problem of turbulence which have been carried on at Göttingen for about five years have unfortunately left the hope of a thorough understanding of turbulent flow very small. The photographs and kinetographic pictures have shown us only how hopelessly complicated this flow is …

Prandtl spoke at a time when numerical calculations made use of primitive devices – slide rules and mechanical desk calculators. We are no longer “hopeless” because DNS provides us with complete details of simple turbulent flows, while experiments have advanced with the help of new techniques including non-obtrusive laser-Doppler and particle-image velocimetry. Also, developments in large-eddy simulation (LES) are also likely to be helpful although this method also involves approximations, both in the filter separating the large (low-wave-number) eddies and the small ‘sub-grid-scale’ eddies, and in the semi-empirical models for the latter.

Even LES is currently too expensive for routine use in engineering, and a common procedure is to adopt the decomposition first introduced by Reynolds for incompressible flows in which the turbulent motion is assumed to comprise the sum of mean (usually time-averaged) and fluctuating parts, the latter covering the whole range of eddy sizes. When introduced into the Navier–Stokes equations in terms of dependent variables the time-averaged equations provide a basis for assumptions for turbulent diffusion terms and, therefore, for attacking mean-flow problems. The resulting equations and their reduced forms contain additional terms, known as the Reynolds stresses and representing turbulent diffusion, so that there are more unknowns than equations. A similar situation arises in transfer of heat and other scalar quantities. In order to proceed further, additional equations for these unknown quantities, or assumptions about the relationship between the unknown quantities and the mean-flow variables, are required. This is referred to as the “closure” problem of turbulence modeling.

The subject of turbulence modeling has advanced considerably in the last forty years, corresponding roughly to the increasing availability of powerful digital computers. The process started with ‘algebraic’ formulations (for example, algebraic formulas for eddy viscosity) and progressed towards methods in which partial differential equations for the transport of turbulence quantities (eddy viscosity, or the Reynolds stresses themselves) are solved simultaneously with reduced forms of the Navier–Stokes equations. At the same time numerical methods have been developed to solve forms of the conservation equations which are more general than the two-dimensional boundary layer equations considered at the Stanford Conference of 1968.

The first edition of this book was written in the period from 1968 to 1973 and was confined to algebraic models for two-dimensional boundary layers. Transport models were in their infancy and were discussed without serious application or evaluation. There were no similar books at that time. This situation has changed and there are several books to which the reader can refer. Books on turbulence include those of Tennekes and Lumley [2], Lesieur [3], Durbin and Petterson [5]. Among those on turbulence models the most comprehensive is probably that of Wilcox [6].

The present book has greater emphasis on modern numerical methods for boundary-layer equations than the first edition and considers turbulence models from advanced algebraic to transport equations but with more emphasis on engineering approaches. A second volume entitled Turbulence Models and Their Application extends this subject to encompass separated flows within the framework of interactive boundary layer theory.

This chapter provides some of the terminology used in subsequent chapters, provides examples of turbulent flows and their complexity, and introduces some important turbulent-flow characteristics.

1.2 Turbulence – Miscellaneous Remarks

We start this chapter by addressing the question “What is turbulence?” In the 25^th Wilbur Wright Memorial Lecture entitled “Turbulence,” von Kármán [7] defined turbulence by quoting G.I. Taylor as follows:

Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid su...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Copyright
• Preface to the Second Edition
• Chapter 1: Introduction
• Chapter 2: Conservation Equations for Compressible Turbulent Flows
• Chapter 3: Boundary-Layer Equations
• Chapter 4: General Behavior of Turbulent Boundary Layers
• Chapter 5: Algebraic Turbulence Models
• Chapter 6: Transport-Equation Turbulence Models
• Chapter 7: Short Cut Methods
• Chapter 8: Differential Methods with Algebraic Turbulence Models
• Chapter 9: Differential Methods with Transport-Equation Turbulence Models
• Chapter 10: Companion Computer Programs
• Subject Index