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
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About the Author
Chair of the Department of Aerospace Engineering, California State University, Professor Cebeci is widely regarded as an expert in the field of Turbulent Flows and has received many accolades for his work. He was named the first Distinguished Professor in the California State University System, and he received numerous awards including Fellow of the American Institute of Aeronautics and Astronautics. He also received the Presidential Science Award from Turkey.
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Analysis of Turbulent Flows with Computer Programs
By Tuncer Cebeci
ElsevierCopyright © 2013 Elsevier Ltd.
All rights reserved.
Chapter Outline Head 1.1 Introductory Remarks 1
1.2 Turbulence – Miscellaneous Remarks 3
1.3 The Ubiquity of Turbulence 7
1.4 The Continuum Hypothesis 8
1.5 Measures of Turbulence – Intensity 11
1.6 Measures of Turbulence – Scale 14
1.7 Measures of Turbulence – The Energy Spectrum 19
1.8 Measures of Turbulence – Intermittency 22
1.9 The Diffusive Nature of Turbulence 23
1.10 Turbulence Simulation 26
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 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 kineto-graphic 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 seventy 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, Lesieur, Durbin and Petterson. Among those on turbulence models the most comprehensive is probably that of Wilcox.
The second edition of this book had greater emphasis on modern numerical methods for boundary-layer equations than the first edition and considered turbulence models from advanced algebraic to transport equations but with more emphasis on engineering approaches. The present edition 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 25th Wilbur Wright Memorial Lecture entitled "Turbulence," von Kármán 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 surfaces or even when neighboring streams of the same fluid flow past or over one another.
That definition is acceptable but is not completely satisfactory. Many irregular flows cannot be considered turbulent. To be turbulent, they must have certain stationary statistical properties analogous to those of fluids when considered on the molecular scale. Hinze recognizes the deficiency in von Kármán's definition and proposes the following:
Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned.
In addition turbulence has a wide range of wave lengths. The three statements taken together define the subject adequately.
What were probably the first observations of turbulent flow in a scientific sense were described by Hagen. He was studying flow of water through round tubes and observed two distinct kinds of flow, which are now known as laminar (or Hagen-Poiseuille) and turbulent. If the flow was laminar as it left the tube, it looked clear like glass; if turbulent, it appeared opaque and frosty. The two kinds of flow can be generated readily by many household faucets. Fifteen years later, in 1854, he published a second paper showing that viscosity as well as velocity influenced the boundary between the two flow regimes. In his work he observed the mean velocity [bar.u] in the tube to be a function of both head and water temperature. (Of course, temperature uniquely determines viscosity.) His results are shown in Fig. 1.1 for several tube diameters. The plot contains implicit variations of [bar.u], r0, and v, the velocity, the tube radius, and the kinematic viscosity, respectively. This form of presentation displays no orderliness in the data. About thirty years later, Reynolds introduced the parameter Rr [equivalent to] [bar.u]r0 /ν an example of what is now known as the Reynolds number (with velocity and length scales depending on the problem). It collapsed Hagen's data into nearly a single curve. The new parameter together with the dimensionless friction factor λ, defined such that the pressure drop Δp = λ([partial derivative][bar.u]2/2) (l/r0), transforms the plot of Fig. 1.1 to that of Fig. 1.2. The quantity l is tube length; the other quantities have the usual meaning. Thus was born the parameter, Reynolds number. The term "turbulent flow" was not used in those earlier studies; the adjective then used was "sinuous" because the path of fluid particles in turbulent flow was observed to be sinusoidal or irregular. The term "turbulent flow" was introduced by Lord Kelvin in 1887.
In the definition of turbulence, it is stated that the flow is irregular. The extreme degree of irregularity is illustrated in Fig. 1.3. If a fine wire is placed transversely in flowing water and given a very short pulse of electric current, electrolysis occurs and the water is marked by minute bubbles of hydrogen that are shed from the length of the wire, provided that the polarity is correct. These bubbles flow along with the stream and mark it. In simple rectilinear flow, the displacement is Δx = uΔt, or, more generally, since u, y, and w motion can occur, Δr = ∫t0 v dτ, where r is the displacement vector, v the velocity vector, and t and s time. Hence the displacement is proportional to the velocity, provided that the times are not too long. The sequence of profiles in Fig. 1.3a was obtained by this hydrogen-bubble technique. All are for the same point in a boundary-layer flow, but at different instants. The variation from instant to instant is dramatic. Figure 1.3b, the result of superposition, shows the time-average displacement for the 17 profiles, and Fig. 1.3c shows the conventional theoretical shape. The average shape remains steady in time, and it is this steadiness of statistical values that makes analysis possible. But Fig. 1.3 shows strikingly that the flow is anything but steady; it is certainly not even a small-perturbation type of flow.
The Reynolds-number parameter has a number of interpretations, but the most fundamental one is that it is a measure of the ratio of inertial forces to viscous forces. It is well known that inertial forces are proportional to [??]V2. Viscous forces are proportional to terms of the type μ[partial derivative]u/[partial derivative]y, or approximately to μV/l, for a given geometry. The ratio of these quantities is
[??]V2/(μ V/l) = [??]Vl/μ [equivalent to] Rl, (1.2.1)
which is a Reynolds number. Whenever a characteristic Reynolds number Rl is high, turbulent flow is likely to occur. In the tube tests of Fig. 1.2, the flow is laminar for all conditions where Rr is below about 1000, and it is turbulent for all conditions where Rr is greater than about 2000. Between those values of Rr is the transition region. Accurate prediction of the transition region is a complicated and essentially unsolved problem.
One fact that is often of some assistance in predicting transition will be mentioned here. Numerous experiments in tube flow with a variety of entrance conditions or degrees of turbulence of the entering flow exist. Preston notes from this information that it seems impossible to obtain fully turbulent flow in a tube at Reynolds numbers Rr less than about 1300 to 2000. His observation is confirmed by the data of Fig. 1.2. Then by considering the similarity of the wall flow for both tube and plate he transfers this observation to low-speed flat-plate flow and concludes that turbulent flow cannot exist below a boundary-layer Reynolds number Rθ [equivalent to] ueθ/ν of about 320, where ue is the edge velocity and q is the momentum thickness defined by
θ = ∫∞0 (1 - u/ue) dy (1.2.2)
If the laminar boundary layer were to grow naturally from the beginning of the flat plate, the x Reynolds number, Rx = uex/ν, would be about 230,000 for Rθ = 320. However, under conditions of very low turbulence in an acoustically treated wind tunnel, an x Reynolds number of 5,000,000 can be reached. Hence, it has been demonstrated that there is a spread ratio of more than 20:1 in which the flow may be either laminar or turbulent. Preston's observation is of importance when turbulent boundary layers are induced by using some sort of roughness to trip the laminar layer, as in wind-tunnel testing. If the model scale is small, Rθ at the trip may be less than 320. Then the trip must be abnormally large – large enough to bring Rθ up to 320. Fortunately, however, the Reynolds number is often so great that there is no problem.
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Table of ContentsTurbulent Boundary Layers; Laminar-Turbulent Transition; Viscous and Inviscid coupling in between the boundary layer; Inviscid Flow; Two-Dimensional and Three-Dimensional incompressible flows; Physical and numerical aspects of Boundary-layer flows; Non-Linear Parabolic Partial Differential Equations