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Unformatted text preview: The Length, Time and Velocity Scales of Turbulence Tony Burden’s Lecture Notes, Spring 2006 These lecture notes are intended to make life easier for the lecturer and the students by reducing the amount of text that is first written on the blackboard and then copied into note books. These notes should be complemented by reading in text books such as “Turbulent Flows” by S.B. Pope or “A First Course in Turbulence” by H. Tennekes and J.L. Lumley. When developing an understanding of turbulence it is a good idea to read in several different books. Literature recommendations are included at the end of each chapter in these notes. The velocity scale of the turbulence In this lecture the velocity scale of the turbulence as a whole is denoted by u T . In general, u T ' K 1 / 2 where K = 1 2 h u i u i i is the mean kinetic energy in the turbulence. Examples of u T can be found in the classical turbulent flows which have been studied earlier in the course. In a turbulent boundary layer u T ' u τ , the friction velocity. More generally in thin shear layers, u 2 T ' ( δ/L ) U Δ U (in the notation of the lecture notes). The length scale of the turbulence In this lecture the length scale of the turbulence as a whole is denoted by l T . Later on in this course, we will be able to say that l T has the same order of magnitude as the integral length scales. Examples of l T can be found in the classical turbulent flows which have been studied earlier in the course. In self-similar wake flows l T ∼ l , the width of the wake, and in a turbulent boundary layer l T ∼ y , the distance to the wall. The Reynolds number of the turbulence is defined to be the dimensionless expression, Re T = u T l T ν = l 2 T /ν l T /u T = viscous time scale convective time scale ≈ rate of change due to inertia etc . rate of change due to viscous stresses example: the earth’s planetary boundary layer l T ∼ 1 km , u T ∼ 1 m / s → Re T ∼ 10 8 1 The energy-bearing eddies An ‘eddy’ is a loose concept that is used when discussing the scales of motion in turbulence and in particular the swirling structures that can be observed in turbulent flow. If an eddy has length scale l and velocity scale u then its time scale is l/u . This is possibly the only firm property of an eddy. In some qualitative discussions the words ‘eddy’ and ‘scale’ are interchangeable. When Re T is large, we can talk about the so called energy–bearing eddies. These are the rather large, most intensive, eddies, that are directly generated by shear in the mean flow. We shall see that the kinetic energy in the turbulence is concentrated to these eddies. Since the energy-bearing eddies dominate the influence of the turbulence on the mean flow the scales u T and l T are used to characterize both these eddies and the turbulence itself....
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.
- Spring '11