Fluid Dynamics

Abstract

This is Part 3 of a book series on fluid dynamics. This is designed to give a comprehensive and coherent description of fluid dynamics, starting with chapters on classical theory suitable for an introductory undergraduate lecture courses, and then progressing through more advanced material up to the level of modern research in the field. This book is devoted to high-Reynolds number flows. It begins by analysing the flows that can be described in the framework of Prandtl’s 1904 classical boundary-layer theory. These analyses include the Blasius boundary layer on a flat plate, the Falkner-Skan solutions for the boundary layer on a wedge surface, and other applications of Prandtl’s theory. It then discusses separated flows, and considers first the so-called ‘self-induced separation’ in supersonic flow that was studied in 1969 by Stewartson and Williams, as well as by Neiland, and led to the ‘triple-deck model’. It also presents Sychev’s 1972 theory of the boundary-layer separation in an incompressible fluid flow past a circular cylinder. It discusses the triple-deck flow near the trailing edge of a flat plate first investigated in 1969 by Stewartson and in 1970 by Messiter. It then considers the incipience of the separation at corner points of the body surface in subsonic and supersonic flows. It concludes by covering the Marginal Separation theory, which represents a special version of the triple-deck theory, and describes the formation and bursting of short separation bubbles at the leading edge of a thin aerofoil.

On boundary-layer receptivity to entropy waves

“…Substitution of (3.3) into the Navier–Stokes equations (2.2) leads to classical boundary-layer equations for compressible flow. A detailed discussion of these equations together with corresponding boundary conditions may be found in § 1.10 in the book by Ruban (2018). We shall assume here that the plate surface is thermally isolated, in which case the boundary-layer equations admit a self-similar solution in the form where Functions , , and are to be found by solving the following set of ordinary differential equations: subject to the boundary conditions …”

“…To describe the behaviour of the solution near the outer edge of the boundary layer we use the procedure suggested in problem 2 in exercises 1 in Ruban (2018). We find that Constants and are found through numerical solution of (3.6) as a whole.…”

“…The equations for the and terms are obtained by substituting (6.5), (6.6 a – c ) into the Navier–Stokes equations (2.2). The perturbations produced by the wall roughness are governed by the linearised Euler equations These equations were encountered on numerous occasions before in the context of the triple-deck theory; see § 4.2.3 in Ruban (2018). They may be reduced, by means of elimination, to a single equation for the pressure With known the enthalpy , longitudinal velocity and density are calculated as The equations for the terms are found to be Performing the elimination routine again we find that the pressure satisfies the following equation: …”

“…These are obtained from the solution of the boundary-layer equations at the position of the roughness Substitution of (6.12) into the Navier–Stokes equations (2.2) yields in the leading-order approximation These are conventional incompressible steady flow boundary-layer equations. They have to be solved with the no-slip conditions on the surface of the roughness (6.1) and the following condition of matching with the solution in the boundary layer upstream of the roughness: where is dimensionless skin friction given by It may be shown (see § 2.2.4 in Ruban 2018) that the solution to (6.15) satisfying condition (6.17) exhibits the following behaviour at the outer edge of the lower tier: where is termed the displacement function.…”

On subsonic boundary-layer receptivity to acoustic waves over an aircraft wing coated by a thin liquid film

“…Therefore triple-deck theory has been receiving a great deal of attention ever since and it has been extended to wide range of problems in fluid dynamics such as the monograph by Sychev et al. (1998) and, in a more recent study, Ruban (2018) focuses on the boundary-layer separation.…”