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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
A guide to the essential information needed to model and compute turbulent flows and interpret experiments and numerical simulations
Turbulent Fluid Flow offers an authoritative resource to the theories and models encountered in the field of turbulent flow. In this book, the author – a noted expert on the subject – creates a complete picture of the essential information needed for engineers and scientists to carry out turbulent flow studies. This important guide puts the focus on the essential aspects of the subject – including modeling, simulation and the interpretation of experimental data - that fit into the basic needs of engineers that work with turbulent flows in technological design and innovation.
Turbulent Fluid Flow offers the basic information that underpins the most recent models and techniques that are currently used to solve turbulent flow challenges. The book provides careful explanations, many supporting figures and detailed mathematical calculations that enable the reader to derive a clear understanding of turbulent fluid flow. This vital resource:
- Offers a clear explanation to the models and techniques currently used to solve turbulent flow problems
- Provides an up-to-date account of recent experimental and numerical studies probing the physics of canonical turbulent flows
- Gives a self-contained treatment of the essential topics in the field of turbulence
- Puts the focus on the connection between the subject matter and the goals of fluids engineering
- Comes with a detailed syllabus and a solutions manual containing MATLAB codes, available on a password-protected companion website
Written for fluids engineers, physicists, applied mathematicians and graduate students in mechanical, aerospace and civil engineering, Turbulent Fluid Flow contains an authoritative resource to the information needed to interpret experiments and carry out turbulent flow studies.
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Table of Contents
Cover
Dedication
Preface
About the Companion Website
1 Introduction
1.1 What is Turbulent Flow?
1.2 Examples of Turbulent Flow
1.3 The Goals of a Turbulent Flow Study
1.4 Overview of the Methodologies Available to Predict Turbulence
1.5 The Plan for this Book
References
2 Describing Turbulence
2.1 Navier–Stokes Equation and Reynolds Number
2.2 What Needs to be Measured and Computed
Reference
Problems
3 Overview of Turbulent Flow Physics and Equations
3.1 The Reynolds Averaged Navier–Stokes Equation
3.2 Turbulent Kinetic Energy Equation
3.3
Equation
3.4 Reynolds Stress Equation
3.5 Vorticity Equation
3.6 Enstrophy Equation
References
Problems
4 Turbulence at Small Scales
4.1 Spectral Representation of
4.2 Consequences of Isotropy
4.3 The Smallest Scales
4.4 Inertial Subrange
4.5 Structure Functions
4.6 Chapter Summary
References
Problems
5 Energy Decay in Isotropic Turbulence
5.1 Energy Decay
5.2 Modes of Isotropic Decay
5.3 Self‐Similarity
5.4 Implications for Turbulence Modeling
5.5 Equation for Two‐Point Correlations
5.6 Self‐Preservation and the Kármán–Howarth Equation
5.7 Energy Spectrum Equation
5.8 Energy Spectrum Equation via Fourier Analysis of the Velocity Field
5.9 Chapter Summary
References
Problems
6 Turbulent Transport and its Modeling
6.1 Molecular Momentum Transport
6.2 Modeling Turbulent Transport by Analogy to Molecular Transport
6.3 Lagrangian Analysis of Turbulent Transport
6.4 Transport Producing Motions
6.5 Gradient Transport
6.6 Homogeneous Shear Flow
6.7 Vorticity Transport
6.8 Chapter Summary
References
Problems
7 Channel and Pipe Flows
7.1 Channel Flow
7.2 Pipe Flow
References
Problems
8 Boundary Layers
8.1 General Properties
8.2 Boundary Layer Growth
8.3 Log‐Law Behavior of the Velocity Mean and Variance
8.4 Outer Layer
8.5 The Structure of Bounded Turbulent Flows
8.6 Near‐Wall Pressure Field
8.7 Chapter Summary
References
Problems
9 Turbulence Modeling
9.1 Types of RANS Models
9.2 Eddy Viscosity Models
9.3 Tools for Model Development
9.4 Non‐Linear Eddy Viscosity Models
9.5 Reynolds Stress Equation Models
9.6 Algebraic Reynolds Stress Models
9.7 URANS
9.8 Chapter Summary
References
Problems
10 Large Eddy Simulations
10.1 Mathematical Basis of LES
10.2 Numerical Considerations
10.3 Subgrid‐Scale Models
10.4 Hybrid LES/RANS Models
10.5 Chapter Summary
References
Problems
11 Properties of Turbulent Free Shear Flows
11.1 Thin Flow Approximation
11.2 Turbulent Wake
11.3 Turbulent Jet
11.4 Turbulent Mixing Layer
11.5 Chapter Summary
References
Problems
12 Calculation of Ground Vehicle Flows
12.1 Ahmed Body
12.2 Realistic Automotive Shapes
12.3 Truck Flows
12.4 Chapter Summary
References
Author Index
Subject Index
End User License Agreement
List of Tables
Chapter 12
Table 12.1 Drag coefficient for the estate and fastback configurations using RAN...
List of Illustrations
Chapter 1
Figure 1.1 Artist's rendering of smoke visualized flow past a tractor trailer....
Figure 1.2 Fog wind tunnel visualization of a NACA 4412 airfoil at a low‐speed ...
Figure 1.3 Clouds forming over Mount Duval, Australia are a visual indicator of...
Figure 1.4 Transition to turbulence in a jet. Courtesy of J.‐L. Balint and L. O...
Figure 1.5 A smoke plume from the Dunbar Cement Works chimney. cc‐by‐sa/2.0 ‐ ©...
Figure 1.6 A picture of the Horn Rev offshore wind farm in Denmark. Turbulence ...
Figure 1.7 Clouds mark the presence of turbulent von Kármán vortex streets form...
Chapter 2
Figure 2.1 (a) Longitudinal and (b) transverse velocities appearing in the defi...
Figure 2.2 Definition of the microscale
.
Chapter 4
Figure 4.1 Spectral ranges of
and
, with
and
marking their respective p...
Figure 4.2 Rotational invariance in isotropic turbulence. The two‐point correla...
Figure 4.3 Antisymmetry of
under reflection. The two‐point correlations based...
Figure 4.4 Definition of two‐point triple velocity correlations: (a)
, (b)
, ...
Figure 4.5 Antisymmetry of the two‐point longitudinal triple velocity correlati...
Figure 4.6 Confirmation of the isotropic identity Eq. ( 4.33 ) from a numerica...
Figure 4.7 Experimental tests of the
law [ 3 ].
Figure 4.8 Compensated energy spectrum as given in [21]. With increasing
the ...
Figure 4.9 Dissipation rate on a plane showing intermittency within a region of...
Figure 4.10 Compensated longitudinal structure functions computed in isotropic ...
Chapter 5
Figure 5.1 Measured and predicted
in the final period [ 2 ]. With permissio...
Figure 5.2 Confirmation of the
decay law in the final period [ 2 ].
,
;
,...
Figure 5.3 Self‐similar decay corresponding to
and
: —,
;
,
.
Figure 5.4 Self‐similar decay corresponding to
and
: —,
;
,
;
,
;
, li...
Figure 5.5
in isotropic decay corresponding to the conditions in Fig. 5.4.
Figure 5.6 Measured power law exponents in decaying homogeneous turbulence from...
Figure 5.7
in the final period.
Figure 5.8 Energy spectrum budget in final period. —,
;
, transfer term;
, d...
Figure 5.9 Triads of wave numbers.
Chapter 6
Figure 6.1 Molecules crossing the surface
in the positive
direction during...
Figure 6.2 A local linear approximation to the mean velocity field
at a point...
Figure 6.3 Ensemble of paths, each with a different initial position
, arrivin...
Figure 6.4 Decomposition in Eq. ( 6.24 ) at
.
,
with zero point,
, denote...
Figure 6.5 Evaluation of Eq. ( 6.24 ) at
computed across the channel. —,
;
Figure 6.6 Contributions to
at
from a data set consisting of 169,344 points...
Figure 6.7 Fraction of points in the data ensembles that account for the local ...
Figure 6.8 Fluid particle arriving at
[7] due to a sweep event. Time increase...
Figure 6.9 Fluid particle arriving at
[ 7 ] due to an ejection event. Time i...
Figure 6.10 Eddy viscosity in channel flow:
,
; —,
.
Figure 6.11 Inadequacy of gradient transport physics: —,
;
,
.
Figure 6.12 A region of negative eddy viscosity can be expected in a channel fl...
Figure 6.13Figure 6.13 Demonstration of the near constancy of
in the central r...
Figure 6.14 Measured
in homogeneous shear flow for
from [15].
Figure 6.15 Computed solution for
(left) and
(right) in homogeneous shear f...
Figure 6.16 Computed solutions for
(left) and
(right) in homogeneous shear ...
Figure 6.17Figure 6.17
:
, DNS results; —, prediction from Eq. ( 6.62 ).
Figure 6.18Figure 6.18
:
, DNS results; —, prediction from Eq. ( 6.63 ).
Figure 6.19Figure 6.19
:
, DNS results; —, prediction from Eq. ( 6.60 ).
Figure 6.20Figure 6.20
:
, DNS results; —, prediction from Eq. ( 6.61 ).
Chapter 7
Figure 7.1 Geometry of channel flow.
Figure 7.2 Average velocity in channel flow of width
scaled by mean centerlin...
Figure 7.3 Decomposition of the total stress as given by Eq. (7.14) in turbulen...
Figure 7.4 Decomposition of the mean momentum equation ( 7.16 ) in turbulent c...
Figure 7.5 Semi‐log plot of
showing an approximate log‐law behavior.
,
;
,...
Figure 7.6
as defined in Eq. (7.32) for the mean velocities in Figure 7.5.
,...
Figure 7.7 Normal Reynolds stresses in channel flow. —,
;
,
. Top curves are...
Figure 7.8 Normal Reynolds stresses in channel flow plotted with respect to
. ...
Figure 7.9 Turbulent kinetic energy budget in channel flow
[ 10 ] scaled wit...
Figure 7.10
equation budget in channel flow at
[ 20 ] scaled with
and
....
Figure 7.11
budget in channel flow for
[ 10 ]. —, production;
, dissipati...
Figure 7.12
budget in channel flow for
[ 10 ].
, dissipation;
, pressure...
Figure 7.13
budget in channel flow for
[ 10 ].
, dissipation;
, pressure...
Figure 7.14 Pressure‐strain term in normal Reynolds stress equations at
[ 10 ...
Figure 7.15
budget in channel flow for
[ 10 ]. —, production;
, dissipati...
Figure 7.16 Comparison of the enstrophy components in channel flow at
[ 10 ]...
Figure 7.17 Evaluation of the terms in Eq. ( 7.54 ) in channel flow with
[ 1...
Figure 7.18 Mean velocity profiles in pipe flow [ 6 ] showing the collective a...
Figure 7.19 Plots of
in pipe flow for empirically fitted exponents,
. From l...
Figure 7.20
vs.
where
in this figure. Data are taken from 16 different Re...
Figure 7.21 Streamwise velocity variance at high Reynolds numbers in pipe flow ...
Chapter 8
Figure 8.1 Turbulent boundary layer over a flat plate.
Figure 8.2 Smoke visualization of a turbulent boundary layer at
[1].
Figure 8.3 Boundary layer zones, not drawn to scale.
Figure 8.4 Intermittency factor in a turbulent boundary layer [ 5 ].
Figure 8.5 Mean and variance of the streamwise velocity in boundary layer flow ...
Figure 8.6 Visualization of the rotational field within
vortices in an H‐type...
Figure 8.7 Turbulent spot visualized using aluminum particles in a transitionin...
Figure 8.8 Contours of streamwise velocity in the boundary layer forming in the...
Figure 8.9 Visualization of low‐speed streaks at
. From [ 28 ]. Reprinted wit...
Figure 8.10 Contour plot of streamwise velocity fluctuation
in a DNS of chann...
Figure 8.11 Visualization of pockets in a smoke‐marked boundary layer. From [33...
Figure 8.12 End‐on velocity vector plots showing vortices in a channel flow. Fr...
Figure 8.13 Instantaneous view of quasi‐streamwise vortices in channel flow wit...
Figure 8.14 Comparison between views with transverse light plane inclined at (a...
Figure 8.15 Visualization of vortices in transition using
. From [50]. Reprint...
Figure 8.16 A lifted and forward sheared vortex line oriented in the
directio...
Figure 8.17 Overhead view of vortex filaments in a transitioning boundary‐layer...
Figure 8.18 During transition, isosurfaces marking low‐speed streaks (dark shad...
Figure 8.19 The vorticity in a lifted‐up furrow adopts a mushroom‐like shape wh...
Figure 8.20 Isosurfaces of
forming what appears to be two nested hairpins tha...
Figure 8.21 Single hairpin representing the rotational signature of a tilted fu...
Figure 8.22 Pockets seen in a numerical simulation of boundary layer flow. From...
Figure 8.23 Filaments at a fixed time showing the progression to turbulent flow...
Figure 8.24 Isosurfaces of
in a fully turbulent boundary layer at
. From [53...
Figure 8.25 Contour plots on a plane intersecting the low‐speed streak in Figur...
Figure 8.26 Contour plots on a plane intersecting the low‐speed streak in Figur...
Figure 8.27 Contour plots on a plane intersecting the low‐speed streak in Figur...
Figure 8.28 Wall pressure maxima generated by coherent vortices. From [ 39 ]. ...
Chapter 9
Figure 9.1 Prediction of the return to isotropy in plane strain. —, SSG model;...
Figure 9.2 Normal components of
during return to isotropy. —, SSG model;
, L...
Figure 9.3 Prediction of
in homogeneous shear flow with
: —, DNS [ 47 ];
, ...
Chapter 10
Figure 10.1 Filters: —, top hat;
, Gaussian;
, sharp Fourier cut‐off.
Chapter 11
Figure 11.1 Basic characteristics of the mean flow in (a) wakes, (b) jets, and...
Figure 11.2 Circular cylinder wake at
; smoke wire at (a)
and (b)
, [ 3 ]....
Figure 11.3 Comparison of the self‐similar turbulent wake velocity profile of a...
Figure 11.4 Centerline mean velocity and jet width development of a turbulent p...
Figure 11.5 Mean streamwise velocity profiles of turbulent plane jet at
for
Figure 11.6 Growth of
along the centerline of a turbulent plane jet at
.
, ...
Figure 11.7 Velocity variances for turbulent plane jet at
and
. Data from [ ...
Figure 11.8 Reynolds shear stress distribution for turbulent plane jet at
and...
Figure 11.9 Growth of a turbulent mixing layer with
, and
denoting the strea...
Figure 11.10 Overhead (top) and side (lower) views of a vortex filament computa...
Figure 11.11 Detail of braid/roller vortices developing in a vortex filament si...
Figure 11.12 Detail of a chain‐link fence vortex pattern developing in a vortex...
Figure 11.13 Overhead view of a mixing layer containing oblique roller/braid vo...
Figure 11.14 Mean streamwise velocity distributions in a self‐preserving two‐st...
Figure 11.15 Normal Reynolds stresses in a two‐stream mixing layer. Here,
. Ex...
Figure 11.16 Reynolds shear stress distributions in a two‐stream mixing layer. ...
Chapter 12
Figure 12.1 Ahmed body. From [4]. Used with permission of Elsevier. (a) Some o...
Figure 12.2 Prediction of the mean velocity for the Ahmed body with rear slant ...
Figure 12.3 Prediction of mean velocity (a and b) and kinetic energy (c and d) ...
Figure 12.4 Prediction of mean velocity for the Ahmed body with rear slant angl...
Figure 12.5 Prediction of mean velocity for the Ahmed body with rear slant angl...
Figure 12.6 Prediction of
for the Ahmed body with rear slant angle
. From [ ...
Figure 12.7 Prediction of mean velocity for the Ahmed body with rear slant angl...
Figure 12.8 Isosurfaces of
for the Ahmed body with
rear slant computed from...
Figure 12.9 Isosurfaces of
for the Ahmed body with
rear slant computed from...
Figure 12.10 Mean streamlines on the central plane of the Ahmed body. From [ 10...
Figure 12.11 DrivAer models. F, fastback; E, estate; N, notchback. From [ 4 ]....
Figure 12.12 Mean pressure coefficient on the top center of the vehicles. From ...
Figure 12.13 Fastback wake structure indicated by the region of reverse flow
....
Figure 12.14 Estate wake structure indicated by the region of reverse flow
. F...
Figure 12.15 Asmo vehicle. From [ 9 ]. Used with permission of Elsevier.
Figure 12.16 Pressure on back surface of the Asmo vehicle.
, VMS/WALE; —, WALE...
Figure 12.17 Pressure on flat back surface of the Asmo vehicle. Left, the Smago...
Figure 12.18 Contours of velocity magnitude on the central plane computed with ...
Figure 12.19 The generic conventional model used in testing turbulence predicti...
Figure 12.20 Mean pressure over the rear surface of the trailer. —, computed on...
Figure 12.21 Instantaneous contours of vorticity magnitude on a plane approxima...
Guide
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Turbulent Fluid Flow
Peter S. Bernard
University of Maryland USA
Copyright
This edition first published 2019
© 2019 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
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Library of Congress Cataloging‐in‐Publication Data applied for
9781119106227
Cover Design: Wiley
Cover Images: © Willyam Bradberry/Shutterstock
Dedication
To my children, Jennifer, Alexander, and Rebecca
Preface
During the time period in which this book was written, there have been significant advances in computation and experimental techniques that suggest that a new era in the long history of turbulent flow study is commencing: one in which answers to many long‐held questions are being found and one in which the capabilities for turbulent flow simulation are beginning to become viable for engineering design and research. As an example, some canonical flows are now being simulated and measured at what appears to be the asymptotic small viscosity (high Reynolds number) range which has previously only been the realm of theoretcal speculation. Moreover, simulations of the flow past geometrically complex shapes such as automobiles that utilize billions of gridpoints are now being attempted. At the same time, as significant as the recent advances in simulation and measurement have been, there remains a great distance between current capabiliites in predicting turbulent flows and a future time when the engineering of real‐world turbulent flows can be done with high accuracy and efficiency as a routine matter. This means that a very large part of turbulent flow prediction today relies on traditional modeling and coarse simulation techniques that have been under continuous development for the last five decades.
In view of where the field of turbulence is today, an introductory book such as this one must give an accounting of both the range of new developments in the field of turbulence as well as a description of modern modeling and simulation techniques together with the many significant and fundamental results that support, motivate, and justify them. With these goals in mind, this book is meant to be a readible presentation of the subject that covers many fundamental and new results with some detail, although it avoids the depth of discussion that can be found in the many articles that are cited as part of the development. In many cases, the latter offer the preferred means of getting to a more comprehensive understanding of specific topics that cannot be reasonably included in a volume such as this. This book will be successful if it prepares students to pursue any number of specialized directions within turbulent flow research or enhances the knowledge and capabilities of engineers who are engaged in predicting turbulent flows via commercial software.
The book is bracketed by chapters in which ground vehicle aerodynamics is used as a means of focussing discussion on the nature of the turbulent flow problem. This includes, in the first chapter, what needs to be determined and what kind of phenomena is to be expected and, in the last chapter, how successful current methodologies are in achieving predictions of turbulent flows. In between these chapters, the line of development considers in turn the fundamental processes whose analysis has led to a variety of models that are incorporated in predictive schemes today. An essential part of these developments is considering what is being discovered in recent times. Specifically, what are the main issues being investigated today that may one day improve prediction techniques and give definitive insights into turbulent flow physics.
This book is the outgrowth of many years teaching graduate courses at the University of Maryland devoted to the theory, physics, and prediction of turbulent flow. I am indebted to the many students along the way who prodded me to keep the subject interesting and whose curiosity helped fuel my own interest. A number of problems are included after many of the chapters in the book. Some of these ask for details omitted in the line of development. Others suggest calculations that can be made via relatively uncomplicated codes (e.g., using MATLAB) that model and simulate turbulence or allow for the analysis of its properties from data sets that are readily available on the internet. Many related problems can be easily devised from extension of the problems given in the book.
I would like to express my appreciation to Neil Ashton, Marc Buffat, Jonathan Morrison, Arsensio Oliva, Richard Owens, Ulrich Rist, Philipp Schlatter, Eric Serre, Kidambi Sreenivas, Makoto Tsubokura, and James Wallace for supplying original figures, artwork, and photographs. I am indebted to Bruce Berger, Pat Collins, and Martin Erinin for reading some chapters from the book and providing useful commentary.
About the Companion Website
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Solutions manual
Scan this QR code to visit the companion website.
1Introduction
1.1 What is Turbulent Flow?
Turbulent flow is ubiquitous in nature and technology. From breaking waves on a beach, to vortical eddies in the atmosphere that shake an airplane in flight, to flow across the hull of a submarine and separating into the ocean, to the flow disturbing the landing of a helicopter on the flight deck of a ship, fluid flows in a turbulent state. Why turbulent flow is so common is equivalent to the question of why flows often depart from a laminar state to become turbulent. The answer to this question can explain the ubiquitousness of turbulent flow and simultaneously provide a useful definition of what is meant by turbulent flow.
It is a fact that in many fluid flows there are sources of perturbation that are persistent and inevitable. These can arise from minute imperfections on boundary surfaces or slight variability in the incoming flow field. It is also conceivable [1, 2] that perturbations of a molecular origin can occur, as in the kinds of organized molecular behavior that leads to Brownian motion. How a fluid flow reacts to the presence of perturbations lies at the heart of whether or not the fluid motion is turbulent or will become turbulent. When perturbations to the velocity field appear in an otherwise laminar flow, the action of viscous forces is to diffuse the local momentum excess associated with the disturbance. Depending on the strength of the perturbation and the effectiveness of the viscous smoothing, the flow disturbances may grow, leading to the appearance of three‐dimensional (3D), non‐steady, disorganized motion that is referred to as turbulence or else be damped leading to the maintenance of laminar flow. The balance in this case is between the inertia of fluid particles in motion and the viscous forces acting on them to regularize the local flow field.
As will be considered more formally in the next chapter, the Reynolds number where and are, respectively, characteristic velocity and length scales, and is the kinematic viscosity, is a measure of the ratio of inertia to viscous forces and so has a large role to play in characterizing whether fluid flow is turbulent or not. For small values of the Reynolds number, internal viscous forces dominate and the flow tends to be laminar, or laminarize if it is not initially so. For high Reynolds numbers viscous smoothing is insufficient to prevent the growth of instabilities, with turbulent flow being the result.
The transition of laminar flow to turbulence can follow a number of different routes depending on the magnitude and nature of the perturbations that are present [3]. For slight perturbations a linear instability is triggered that in a boundary layer, for example, would be manifest as a pattern of streamwise disturbances known as Tollmein–Schlichting waves. As they develop downstream vortical structures appear in the flow whose breakdown and interactions signal the appearance of turbulent flow. If the initial perturbation to a laminar flow is sufficiently large, then bypass transition may occur for which there is a rapid development of the vortical structures leading to turbulence. Some aspects of the vortical structures occurring in transition and turbulent flow are discussed in Chapter 8.
Once initiated, turbulent flow persists unless there is a change in external conditions that could remove or reduce the mechanisms leading to instability. For example, flow acceleration in some circumstances has been observed to relaminarize the flow in boundary layers [4]. In many situations, such as pipe and channel flows, and boundary and mixing layers that will be discussed in this book, the flows may be seen to evolve from an upstream laminar state, through transition to the fully turbulent state downstream.
The subject of turbulence is primarily concerned with describing and predicting the quantitative and qualitative properties of fluid flow specifically in turbulent flow regions. The study of the conditions that might lead to the appearance of turbulent flow is the main interest of stability theory. Some overlap between these fields can be expected since the conditions leading to flow breakdown in transition may have some role in the fully turbulent region as well [5]. Moreover, turbulent flow may contain structural features in the form of vortices that also populate the transitional flow region [6]. Studying such aspects of transition can have benefit for the study of turbulent flow as well. For those flows where laminar, transitional, and turbulent flow exist simultaneously, as in a developing boundary layer, it can be important to predict the extent and properties of each separate region.
It will be seen subsequently that there are numerous ways that turbulent fluid motion differs from laminar motion, and this has important consequences for such aspects of flow analysis as predicting the forces on bodies or the diffusion of contaminants within the flow field. Moreover, the special properties of turbulent motion make it much more difficult than laminar flow to either solve for the fluid velocity field via numerical simulations or measure it in physical experiments. It will be seen that limitations of a variety of sorts shadow the analysis of turbulent flow so that in the study of any particular flow decisions often have to be made as to what methods ought to be brought to bear in studying the flow and how and to what extent they should be deployed. The combination of an important need to predict turbulent flow behavior and the fact that analysis and measurement techniques are not without limitations leads to the need for fluids engineers to acquire some degree of proficiency in understanding the advantages and disadvantages of available techniques and how best to interpret what they say about the flow field.
1.2 Examples of Turbulent Flow
Considering some specific examples where turbulent flow is present and important can help make clear the wide range of situations that are encompassed in the study of turbulence. Among the important categories of phenomena involving turbulent motion are those associated with the flow adjacent to solid surfaces. In the near‐wall field lies the origin of the viscous and pressure forces that affect the motion of bodies. Turbulence produced in boundary layers is a common occurrence that has significant consequences for the strength and nature of the boundary forces. Turbulent flow is an integral part of the movement of automobiles and trucks, as seen in Figure 1.1 where knowledge from wind tunnel testing and simulations has been used to recreate the kind of smoke pattern to be expected in the flow around semi‐tractor trailers. Turbulence develops over the trailer as boundary layers that separate into a turbulent wake that has a large influence on the overall drag force. Complicated turbulent eddying appears in the gap between the cab and trailer, and in the underbody with considerable consequences for drag and stability.
Turbulence produced on the wings and fuselage of an aircraft must be taken into account in the design process. For a wing at high angle of attack, as seen in Figure 1.2, turbulent vortices filling the wake and shedding from the edges of the wing are inextricably tied to the dynamics of the airplane by their effect on the pressure field. Similarly, turbulent flow is an important part of producing drag and lift forces affecting the hulls and keels of boats and such objects as runners, sky divers, skiers, ice skaters, and projectiles including baseballs, footballs, and golf balls.
Figure 1.1 Artist's rendering of smoke visualized flow past a tractor trailer. The main areas affected by turbulence include boundary layers on the trailer surface, the tractor‐trailer gap, under the chassis, and the large area of turbulence behind the vehicle. Courtesy of Don‐Bur (Bodies and Trailers) Ltd.
Figure 1.2 Fog wind tunnel visualization of a NACA 4412 airfoil at a low‐speed flow (Re = 20,000). Turbulence fills the massively separated flow on the back of the airfoil and vortices roll up in the trailing edge wake. Image by Georgepehli, Smart Blade GmbH.
Turbulent flow next to walls is also an essential part of the forces in internal flows such as channels and pipes. For low Reynolds numbers, the flow in a pipe will be laminar, that is Poiseulle flow, but beyond a transition Reynolds number depending on such factors as the smoothness of the boundaries and incoming flow, the motion in the pipe will be turbulent. In a smooth‐walled pipe transition occurs at a Reynolds number based on mean flow velocity and diameter of approximately 2000 [7], a condition that is often exceeded in engineering applications. Under some circumstances, such as the presence of blockages, turbulent flow occurs in the human lung and heart. Turbulence is an essential aspect of the combustion process in engines, and the flow through heat exchangers, turbines, and numerous other devices.
Oftentimes, turbulent flows occur away from the immediate effects of solid boundaries in what are known as free shear flows. In the case of mixing layers two streams of differing velocity come together, leading to the development, amplification, and merging of vortical structures that have arisen via Kelvin–Helmholtz instability. An example of this is illustrated in Figure 1.3 where clouds mark the presence of a mixing layer formed from the presence of high wind shear aloft in the atmosphere.
Figure 1.3 Clouds forming over Mount Duval, Australia are a visual indicator of vortices formed within a turbulent mixing layer produced by strong wind shear. Photograph by GRAHAMUK at Wikimedia.
Jet flows under many circumstances develop a strong turbulent field promoted by instability at their interface with surrounding fluid leading to the development and breakdown of vortical structures. This is seen in the jet shown in Figure 1.4 and in the jet propelled by buoyancy forces that is formed from the venting of hot air into the atmosphere from a smokestack, as shown in Figure 1.5. In these images turbulent vortical structures are seen to develop at the base of the jet where it departs the orifice and grow in size downstream until the entire exiting volume of fluid is in turbulent motion.
Figure 1.4 Transition to turbulence in a jet. Courtesy of J.‐L. Balint and L. Ong.
Figure 1.5 A smoke plume from the Dunbar Cement Works chimney. cc‐by‐sa/2.0 ‐ ©Walter Baxter ‐ geograph.org.uk/p/3765299.
Covering a large range of scales, turbulent motion is often found extending long distances downstream in the wakes of flows past objects, as is seen in the wind farm shown in Figure 1.6. The performance of downstream wind turbines is significantly altered by being exposed to the long turbulent wakes of upstream wind turbines. Another example of turbulent wake flows is found in the von Kármán vortex streets shown in Figure 1.7 that form as wind blows past some appropriately shaped and sized Canary Islands. Breakdown of the wake vortices into turbulence is the end stage of the upstream perturbation to the oncoming flow field.
Figure 1.6 A picture of the Horn Rev offshore wind farm in Denmark. Turbulence appearing behind the wind turbines is marked by moisture condensation from the atmosphere. Photo by Christian Steiness. Original image link: http://i.imgur.com/qruVcnu.jpg.
Figure 1.7 Clouds mark the presence of turbulent von Kármán vortex streets forming in the wake of the Canary Islands. NASA image by Jeff Schmaltz, LANCE/EOSDIS Rapid Response.
In many applications turbulence per se represents just one aspect of the essential physics which might also include sound and shock waves, combustion, chemical reactions, natural convection, two‐phase gas–liquid flows as well as flows with particulates, sedimentation and slurries, and electromagnetic phenomena as in plasmas. The presence of turbulence in such circumstances can have a profound influence on phenomena associated with the additional physics. In many cases there is a two‐way coupling in which turbulence both affects and is affected by the presence of other physical phenomena. For example, liquid–gas flow in pipes can take on a number of different regimes depending on the turbulent interaction between phases.
For the many flows where turbulent motion is present it is natural to wonder how its presence might affect the way in which flows are studied. The next section gives some insight into this question by considering in some detail the kinds of turbulent flow phenomena that are present in a particular application, specifically the flow past a semi‐tractor trailer truck. Such flows are of great practical interest and harbor a variety of phenomena that are shared by many other common flows of interest. After discussing what one would like to know about turbulence in such a practical setting, we then describe what sorts of methodologies might be applied to acquire the desired information. Presenting some of the essential knowledge associated with each of these approaches toward studying turbulent flow phenomena then forms the main goal of this book.
1.3 The Goals of a Turbulent Flow Study
The flow past a semi‐tractor trailer truck is complex and includes a variety of phenomena that are likely to be worthy of investigation and analysis to fluids engineers. For example, traveling at 65 mph a typical truck might use approximately 30 of its fuel consumption to overcome aerodynamic drag [8]. Multiplied over the entire population of trucks traveling on any given day, modifications to trucks that reduces drag can provide a very significant environmental and economic gain. Other aspects of truck flows that are subjects of investigation include stability to cross winds, noise generation, splash and spray of road water, and fouling of the windshield from airborne particulates. In all these aspects, turbulent flow plays an integral part and it is important to have a capacity for predicting and understanding its action.
One relatively straightforward aspect of a truck flow consists of the boundary layers that form on the trailers' flat surfaces. For example, on windless days or if the truck is moving head‐on directly into the wind, boundary layers will develop down the length of the upper surface and sides of the trailer. As illustrated in Figure 1.1, the boundary layer on the top surface is turbulent and develops in response to external conditions associated with the free stream velocity and the near‐wall pressure field associated with the truck movement. The initial development of the boundary layer is strongly influenced by the upstream flow conditions including buffeting from fluid separating from the cab and gap as well as complex geometrical features of the of the leading edge of the trailer. In a laboratory setting with laminar upstream flow conditions, transition generally occurs when the Reynolds number . For the circumstances similar to that of a truck traveling at 105 kph (65 mph) in air with /s, the downstream distance in a zero‐pressure gradient boundary layer where transition occurs is cm (8 in). Thus, even in the best of circumstances when a laminar boundary layer is able to form, the non‐turbulent part of the boundary layer will be of little consequence to a typical trailer with length up to 16 m (53 ft). For the flow to be laminar over the truck, it would have to travel at 1 mph. Boundary layers in turbulent flow have been studied extensively and a considerable amount is known about their statistics and properties. Much of this will be considered in subsequent chapters.
At the rear of the truck the boundary layers separate off into a wake region that forms behind the blunt shape of the trailer, as seen in Figure 1.1. The flow here is unlike the sleek aerodynamic wake behind a wing and is not amenable to simple analysis. Vortical eddies produced in the upstream boundary layers join with strong vortical recirculation regions that develop as an outgrowth of separation of the turbulent boundary layer off the rear edge. Out of this complicated non‐steady flow pattern a pressure field is established whose amplitude fails to balance the high pressure on the front surfaces of the cab and trailer. The result is a significant net contribution to the drag force. The problem of predicting drag on trucks is thus in part closely associated with predicting the details of the non‐steady wake flow. This generally means resolving the time history of vortical features in the wake in the course of simulating the flow field.
The truck cab presents itself to the oncoming flow as a geometrically complex bluff body, with the flow over the front surfaces having the character of a potential flow. The wake flow behind the cab is similar to the rear of the trailer and can be expected to produce turbulent upstream conditions for the trailer flow. Depending on whether fairing is placed to cover the gap between cab and trailer, there may be a strong cavity flow in the region between them. The pressure field on the rear of the cab and front of the trailer will affect the overall drag prediction.
The direction and magnitude of the drag on a truck depend strongly on the relative strengths and directions of the ambient wind in relation to the velocity of the truck. In effect, the relative wind as seen by an observer traveling with the truck is the essential quantity needed to determine the drag. The relative velocity is liable to point in any direction. When it is not head‐on there is potential for recirculating flows to form along the sides of the trailer, including significant separation. Such phenomena are associated with large transients in the strength and orientation of side forces that can affect the stability of trucks, particularly when they pass each other.
Specialized aspects of the truck flow that may also be of interest include the noise producing flow around the side mirrors, the flow past the moving tires, the cooling effect of air traveling through the engine compartment, and flow in the irregular underbody region of varying cross‐sectional area. Non‐steadiness of many aspects of the flow imply that capturing the movement and history of the turbulent field is essential for reproducing the physics of such flows. Finally, it is certain that the flow within the engine is mostly turbulent, compressible, involves combustion, two phases, atomizing liquid jets and large gradients in temperature, and other flow properties.
It is evident from this discussion that ideally one would like to have accurate knowledge of the time‐dependent velocity field and pressure for the entire flow past a truck. Out of such data a means is provided for obtaining the complete transient forces and moments acting on the truck. In particular, the velocity field enables computation of the surface shear stress that combines with the pressure to create the force field on the body of the truck. From this the drag and moments acting on the truck can be computed to help determine projected fuel efficiency and stability. Knowing the entire flow also allows for prediction of such practical features as the paths of airborne particles that may interact with the windshield, the calculation of noise generated by the truck, and the dispersion of exhaust gases and particulates into the wake. A capability of simulating the turbulent engine flow can be important for achieving additional gains in minimizing pollution and maximizing engine efficiency. Current capabilities for acquiring the desired data fall short of the ideal and result in the need to make decisions as to what kind of information can be obtained and at what accuracy.
1.4 Overview of the Methodologies Available to Predict Turbulence
Assuming that one is interested in studying a particular turbulent flow, such as the truck flow described in the previous section, decisions have to be made as to what route one should take to best analyze the flow field. As it happens, it is often not obvious what the best strategy should be, and, in fact, there are a variety of different approaches including physical experiments, computation, and modeling that can be pursued with no one of them dependably giving a complete and satisfactory answer to questions about all flow fields. Here, as a preamble to what will be the main preoccupation of the book, we consider the advantages and disadvantages of various strategies for predicting turbulent truck flow as an example of what might occur in the general case.
1.4.1 Direct Numerical Simulation
Since knowledge of the complete transient flow field past a truck is tantamount to knowing all the forces acting everywhere on the truck at all times, any method that can provide this data would be an ideal engineering tool. The potential for acquiring such data numerically depends on the capabilities of schemes for solving the Navier–Stokes equations on a computational mesh surrounding the truck. If successful, such a numerical solution would supply the surface shear stress and pressure field so that forces and moments could be computed as well as other much‐needed information. With an accurate numerical approach physical experiments are unnecessary and parametric studies are relatively easy to accomplish.
The kind of numerical calculation that has just been described is known as a direct numerical simulation (DNS) in the sense that the simulation is performed directly without the use of simplifying steps. Computer storage and speed became sufficiently large in the late 1980s to enable the first DNS of wall‐bounded turbulent flows [9]. For the most part early simulations of shear flows were limited to the study of fully developed channel flow at Reynolds numbers just beyond the transition region. In subsequent years there have been dramatic advances in all aspects of computer technology and as a result great improvements in geometrical complexity and Reynolds numbers in the turbulent flows that can be well simulated.
For a DNS to be appropriate for truck flows it is necessary that the grid resolution in space and time be sufficiently fine to capture the smallest spatial or transient variations in the velocity field. As will be seen in later chapters, turbulent flow contains dissipative processes depending on viscosity that occur over very small distances in comparison to the scale of the overall flow field. The need to resolve such motions creates the need for large meshes that drive up the cost of simulations beyond the point where they are practical using modern supercomputers.
However, while DNS of industrial flows may be precluded because of affordability issues, nonetheless DNS is feasible for a variety of laboratory flows and thus has been and continues to be a great source of knowledge about the physics and modeling of turbulent flow. From such insights come better means of predicting and understanding more complex flows such as those with trucks. DNS solutions can also give insights into how best to develop predictive schemes that compensate for reduced grid resolution by incorporating turbulence models. This is the field of turbulence modeling that represents an alternative to DNS and physical experiments.
1.4.2 Experimental Methods
Since DNS is not feasible for studying complex turbulent flows such as that past a truck, it is necessary to consider alternative approaches. Performing physical experiments in such cases has strong appeal, since real flows at high Reynolds number can be used as the subject of the study. Indeed, experimental methods are used today in a significant capacity in fluids engineering and it is interesting to consider what such studies might offer.
While the ultimate goal of an experimental study of truck flow might be to measure the flow past a full‐sized truck under typical road conditions, it is rarely if ever possible to achieve such measurements in practice. For a start, facilities such as wind tunnels that are large enough to accommodate trucks [10] are rare. More typical is the use of scale‐model trucks in wind‐tunnel studies [11]. In this case there is likely to be some difficulty in establishing complete dynamical similarity between the model and the prototype, for example in maintaining the same Reynolds number. It is also the case that the drag on a truck is normally experienced as it moves over a stationary surface. In wind tunnel studies, the truck is stationary and so to mimic true road conditions a moving floor should be used. Such a capability is not available in all facilities and drag measurements in such instances will almost certainly be affected.
Measurements in wind tunnels are also likely to reflect blockage effects in which the presence of the model reduces the cross‐sectional flow area, leading to a distortion of the flow exterior to the boundary layers forming on the truck surfaces. It is also the case that not all wind tunnels can accommodate the full range of external boundary conditions of interest. For example, imposing side flows requires turning the model, which can present difficulties in many tunnels unless the models are at a sufficiently small scale. It can be expected that capturing transient effects associated with stability will also be difficult to achieve in a wind tunnel experiment.
Independent of the particular flow configuration that is to be studied, decisions have to be made concerning what quantities will be measured and where in the flow field they will be obtained. Thus, sampling of the 3D velocity field around an object such as a truck by measurement techniques is not practical beyond a relatively small number of discrete points. In fact, there is no easy way to capture experimentally the full range of velocity and pressure data that could be obtained from a DNS if one were feasible. To measure the force distribution over the truck surface, flow measurements in its vicinity must be made but this is only doable on a small subset of the total surface, for example by outfitting the model with a finite number of pressure taps.
One special strength of physical measurements is that they allow determination of the integrated effect of the forces on the truck. For example, the drag on a truck or a truck model can be measured through the use of a force balance that records the entire net force and moments on the object to which it is connected. Generally, such data represent a time‐integrated mean force, which is all that is necessary to obtain the drag. Other integrated methodologies for determining drag consist of taking measurements of the average velocity over a 2D plane in the wake of the body. This can be done by sweeping out the area via a rake of probes. The connection of such data to drag arises from theoretical considerations tying the momentum deficit in the wake to the drag force [12].
Flow visualization methods are also useful in studying turbulent flows and in particular the flow past ground vehicles. In this case tufts placed at convenient locations on the surface of the vehicle can give an indication of where separation occurs. Similarly, oil covering the surface of the truck can be used to get a qualitative idea of the location of structural flow features on the surfaces such as separation lines. Another possibility is to introduce smoke or other marker particles to visualize structural aspects of the flow around the truck, as is illustrated in Figure 1.1.
The engineering of turbulent flow may often involve parametric studies as in determining the effect of geometric modifications on drag forces, noise generation, or other flow properties. For an experimental program such tests can require costly fabrication of multiple forms of the model vehicle. For example, for clay models they may need to be resculpted many times to test different shapes. For this reason, as well as those discussed above, it is evident that while experimental measurement programs can provide much needed information they are likely to be done in concert with alternative analyses to get the full range of data that is required in engineering studies. Taken together with the limited applicability of DNS it becomes necessary to consider a variety of approximate means of analyzing turbulent flow that collectively form a wide range of turbulence models. Such approaches allow for numerical analyses of flows aided by insights taken from physical experiments and DNS.
1.4.3 Turbulence Modeling
Turbulence modeling in many guises attempts to fill the gap between the requirements of turbulent flow prediction and the limitations of DNS and experimental methods in satisfying them. As we will see, turbulence modeling most often comes with the penalty of lost accuracy while nonetheless still supplying information about flows that is useful in a practical sense. In other words, by not expecting an exact solution, but only an approximate one, more room is created for developing alternative methodologies for analyzing turbulent flows. Turbulence modeling attempts to use theoretical and empirical results about the behavior of turbulent flow to develop equations whose solutions to varying degrees approximate the actual flow fields.
The most widespread methodology for modeling turbulent flow is based on solving the Reynolds‐averaged Navier–Stokes (RANS) equations that govern the dynamics of the mean velocity field. In this case the averaging is done to the equations themselves prior to solving them, in contrast to DNS where averages are obtained only after the Navier–Stokes equations are solved. Of increasing popularity in recent years as computing resources have grown in size and speed are large eddy simulations (LES) that provide approximations to turbulent flow fields that are simplified by not modeling the detailed motion of small‐scale phenomena. Such effects appear instead in the guise of models of one sort or another. In a more recent trend, a number of hybrid methods have been developed that attempt to combine the best aspects of LES and RANS modeling into one self‐contained approach. Exactly how the modeling is done in RANS, LES, and hybrid LES/RANS and whether or not it can be justified consumes a large part of this book.
It is difficult to predict a priori how well a particular turbulence model will succeed when applied in complex flow situations such as that associated with a truck. Whether or not such calculations are accurate depends on the particular statistic that is being computed and what the criterion for “accuracy” might happen to be. For example, for many popular models the drag might be computed reasonably well, say to within of measured values, yet errors in the underlying pressure distribution may nonetheless be quite significant. In fact, offsetting errors in the pressure prediction computed over different portions of a body may allow for the net pressure force to fall near measured values. How one reacts to this situation depends on the goals of the computational study. Inaccuracies in modeling turbulent flow at the rear of a truck may mean that the wake vortex structure is incorrect, but this may not be of concern if designing the shape of upstream features such as the side mirror is the focus of the study.
Decisions as to whether to pursue RANS vs. LES depend partly on the availability of computer resources. A RANS calculation for the truck flow can be a formidable computation because of the size and complexity of the geometry of the flow. It is likely, however, to be much less expensive than an equivalent LES computation, but whether or not there is value in pursuing LES might depend on the likelihood and importance of achieving better results and at what cost.
Affecting the choice of RANS or LES is also whether or not transient information is needed, as it would be in predicting shed vortices linked to sound generation or investigating stability in the face of sudden side winds. Generally, RANS is less adept at providing transient data than LES so the latter may be the only option. Conversely, for parametric studies requiring many similar calculations, the fact that RANS solutions can be obtained in less time than LES may be a deciding factor in the decision to employ RANS.
Within the RANS and LES approaches, there are numerous modeling choices to consider. Some of the more commonly found methodologies will be described in the following, though this is not to say that other choices in the literature might provide some improvements for specific applications. Indeed, it is the pursuit of better performance that has caused the fracturing of what were originally a small number of modeling approaches into the great number there are today.
1.5 The Plan for this Book
The next chapter considers the kinds of quantities that one might want to determine or measure in a turbulent flow. This is followed by a presentation of the basic equations of motion that can be used as a framework for later discussions of the physics of turbulence. Starting with Chapter 4 and continuing through Chapter 8, we consider in turn the major aspects of the turbulent flow physics whose modeling is the goal of the various methodologies that have been developed for predicting turbulent motion. It will be seen that the most commonly employed turbulence models, whether RANS or LES, are based upon a number of basic notions about the physical nature of turbulent flow that in one way or another can be expressed mathematically to yield a set of equations amenable to numerical solution. The discussion combines empirical and theoretical evidence for the way in which turbulent flow behaves with consideration of how this understanding of the physics finds its way into models.
After having discussed the relationship between the major aspects of turbulence physics and its modeling, Chapter 9 considers the complete RANS models that are built from these analyses. Then Chapter 10 considers LES and hybrid LES/RANS models. Chapter 11
