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Unformatted text preview: LAMINAR BOUNDARY-LAYER THEORY: A 20TH CENTURY PARADOX? Stephen J. Cowley DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK. [email protected] Keywords: Boundary layer, shear layer, separation, singularity, instability. Abstract Boundary-layer theory is crucial in understanding why certain phenom- ena occur. We start by reviewing steady and unsteady separation from the viewpoint of classical non-interactive boundary-layer theory. Next, interactive boundary-layer theory is introduced in the context of un- steady separation. This discussion leads onto a consideration of large- Reynolds-number asymptotic instability theory. We emphasise that a key aspect of boundary-layer theory is the development of singularities in solutions of the governing equations. This feature, when combined with the pervasiveness of instabilities, often forces smaller and smaller scales to be considered. Such a cascade of scales can limit the quan- titative usefulness of solutions. We also note that classical boundary- layer theory may not always be the large-Reynolds-number limit of the Navier-Stokes equations. This is because of the possible amplification of short-scale modes, which are initially exponentially small, by a Rayleigh instability mechanism. 1. INTRODUCTION Sectional lecturers were invited ‘to weave in a bit more retrospective and/or prospective material [than normal] given the particular [Millen- nium] year of the Congress’. This invitation is reflected in the current, possibly idiosyncratic, article. For alternative viewpoints the reader is referred to Stewartson (1981), Smith (1982), Cowley and Wu (1994), Goldstein (1995) and Sychev et al. (1998). We begin with a deconstruc- tion of the components of the title. Boundary-Layer Theory. Prandtl (1904) proposed that viscous effects would be confined to thin shear layers adjacent to boundaries in the case 1 2 of the ‘motion of fluids with very little viscosity’, i.e. in the case of flows for which the characteristic Reynolds number, Re , is large. In a more general sense we will use ‘boundary-layer theory’ (BLT) to refer to any large-Reynolds-number, Re 1, asymptotic theory in which there are thin shear layers (whether or not there are boundaries). 20 th Century. Prandtl (1904) published his seminal paper on the foundations of boundary-layer theory at the start of the 20th century, while the ICTAM 2000 was held at the end of the same century. Laminar. Like Prandtl (1904) we will be concerned with laminar, rather than turbulent, flows. Flows that are in the process of laminar- turbulent transition will be viewed as unstable laminar flows. A Paradox. Experimental flows at large Reynolds numbers are tur- bulent, yet useful comparisons with laminar-flow experiments at mod- erately large Reynolds numbers can sometimes be made with large- Reynolds-number asymptotic theories. We view as a paradox this seem- ingly contradictory result, i.e. that useful comparisons with laminar flow can be made with expansions made about Reynolds numbers when flows...
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This note was uploaded on 01/05/2011 for the course CSU 3 taught by Professor Handsome during the Spring '10 term at CSU Pueblo.
- Spring '10
- The Land