Viscous flow and boundary layers[35]

Preamble

This chapter introduces the concept of the boundary layer, describes the flow phenomena involved, and explains how the Navier Stokes equations can be simplified for the analysis of boundary-layer flows. Certain useful solutions to the boundary-layer equations are described. There are two phenomena, governed by viscous effects and the behaviour of the boundary layer, that are vitally important for engineering applications of aerodynamics. These are How separation and transition from laminar to turbulent flow. A section is devoted to each in turn. The momentum-integral form of the boundary-layer equations is derived. Its use for obtaining approximate solutions for laminar, turbulent, and mixed laminar-turbulent boundary layers is explored in detail. Its application for estimating profile drag is also described. These approximate techniques are illustrated with examples chosen to show how to estimate the aerodynamic characteristics, such as drag, that depend on the behaviour of the boundary layer. Computational methods for obtaining numerical solutions to the boundary-layer equations are presented and reviewed. Some of the computational methods are explained in detail. A substantial section is devoted to the flow physics of turbulent boundary layers with illustrations from aeronautical applications. Computational methods for turbulent boundary layers and other flows are also reviewed. As viscosity is the key physical property governing the behaviour of boundary layers and related phenomena, a quantitative treatment of compressible effects is omitted. But the chapter closes with a detailed qualitative description of the influence of compressible effects on boundary layers, particularly their interaction with shock waves.

7.1 Introduction

In the other chapters of this book, the effects of viscosity, which is an inherent property of any real fluid, have, in the main, been ignored. At first sight, it would seem to be a waste of time to study inviscid fluid flow when all practical fluid

Effects of viscosity negligible in regions not In close proximity to the body

problems involve viscous action. The purpose behind this study by engineers dates back to the beginning of the previous century (1904) when Prandtl conceived the idea of the boundary layer.

In order to appreciate this concept, consider the flow of a fluid past a body of reasonably slender form (Fig. 7.1). In aerodynamics, almost invariably, the fluid viscosity is relatively small (i. e. the Reynolds number is high); so that, unless the transverse velocity gradients are appreciable, the shearing stresses developed [given by Newton’s equation r = ц(ди/ду) (see, for example, Section 1.2.6 and Eqn (2.86))] will be very small. Studies of flows, such as that indicated in Fig. 7.1, show that the transverse velocity gradients are usually negligibly small throughout the flow field except for thin layers of fluid immediately adjacent to the solid boundaries. Within these boundary layers, however, large shearing velocities are produced with conse­quent shearing stresses of appreciable magnitude.

Consideration of the intermolecular forces between solids and fluids leads to the assumption that at the boundary between a solid and a fluid (other than a rarefied gas) there is a condition of no slip. In other words, the relative velocity of the fluid tangential to the surface is everywhere zero. Since the mainstream velocity at a small distance from the surface may be considerable, it is evident that appreciable shearing velocity gradients may exist within this boundary region.

Prandtl pointed out that these boundary layers were usually very thin, provided that the body was of streamline form, at a moderate angle of incidence to the flow and that the flow Reynolds number was sufficiently large; so that, as a first approximation, their presence might be ignored in order to estimate the pressure field produced about the body. For aerofoil shapes, this pressure field is, in fact, only slightly modified by the boundary-layer flow, since almost the entire lifting force is produced by normal pressures at the aerofoil surface, it is possible to develop theories for the evaluation of the lift force by consideration of the flow field outside the boundary layers, where the flow is essentially invisdd in behaviour. Herein lies the importance of the inviscid flow methods considered previously. As has been noted in Section 4.1, however, no drag force, other than induced drag, ever results from these theories. The drag force is mainly due to shearing stresses at the body surface (see Section 1.5.5) and it is in the estimation of these that the study of boundary-layer behaviour is essential.

The enormous simplification in the study of the whole problem, which follows from Prandtl’s boundary-layer concept, is that the equations of viscous motion need

be considered only in the limited regions of the boundary layers, where appreciable simplifying assumptions can reasonably be made. This was the major single impetus to the rapid advance in aerodynamic theory that took place in the first half of the twentieth century. However, in spite of this simplification, the prediction of boundary – layer behaviour is by no means simple. Modern methods of computational fluid dynamics provide powerful tools for predicting boundary-layer behaviour. However, these methods are only accessible to specialists; it still remains essential to study boundary layers in a more fundamental way to gain insight into their behaviour and influence on the flow field as a whole. To begin with, we will consider the general physical behaviour of boundary layers.