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Flow‐induced forces on a body in a viscous fluid arise from:

1 A tangential stress exerted on the surface, the skin friction, which is caused directly by the viscosity in the fluid coupled with the fact that there cannot be any relative motion of a viscous fluid with respect to the body at its surface, the no‐slip condition.

2 A normal stress exerted at the surface, the pressure.

Both types of stress contribute to the drag.

It is convenient to consider the drag exerted on 2‐D bodies across a uniform flow, because many general practical bodies are of a configuration that has one long cross‐flow dimension such that the flow varies only gradually in that ‘long’ direction. In such cases, 2‐D flow is a good local approximation to the flow about any section of the body normal to the long axis. These configurations may be termed quasi‐2‐D. Wind turbine blades and towers are examples of such bodies.

All fluids (with a very few special exceptions, such as liquid helium) have some viscosity, although in the case of two of the most common fluids, air and water, it is relatively small. In the absence of any viscous effect, the flow slips relative to the body at its surface, can be described by a potential function, and is called potential flow. The drag in this case on a 2‐D body in fully subsonic, steady inviscid flow is exactly zero because no wake is generated.

The action of viscosity is to diffuse vorticity and hence momentum in a way analogous to the diffusion of heat, out from the body surface where the flow is retarded by the no‐slip condition, which now applies at the surface. If the fluid has small kinematic viscosity and a comparatively large length scale and velocity so that the Reynolds number is high, the viscous diffusion effects spread outwards at a very much slower speed than the main velocity convection speed along the body surface and as a result remain confined to a thin layer adjacent to the body surface, the boundary layer.

Figure A3.1 Flow past a streamlined body.

Generally, bodies are subdivided into two categories: streamlined and bluff. The main characteristics of streamlined bodies (see, e.g. Figure A3.1) are that the boundary layers remain thin over the whole body surface to the rearmost part of the body, where they recombine and stream off in a thin wake and the drag coefficient is comparatively small. Bodies such as wings and rotor blades whose sections are aerofoils are examples. Bodies on which not all of the boundary layers remain attached in this way up to a trailing edge but rather detach at earlier points creating a thick wake are termed bluff bodies. The flows around such bodies, for example, circular cylinders and fully stalled aerofoils (see Figure A3.12), result in a comparatively high drag coefficient. More general 3‐D bodies may also belong to another category, that of slender bodies (slender in the flow direction), which are not relevant here.

Many practical bodies such as wind turbines or aircraft involve a complex assembly of components that individually belong to the preceding categories. Forces on such bodies are usually calculated by breaking the body down into quasi‐2‐D elements, and interactions between elements are dealt with, when significant, by interference coefficients. In some cases where it is appropriate to consider sectional flow, such as for the blades of a wind turbine, the flow is not exactly in the plane of the section and may contain a non‐zero ‘lengthwise’ or transverse component. It is usual and can be demonstrated that if boundary layer effects are neglected, the pressures and forces on any body section normal to the long axis result from only those flow components that are in the plane of the section and are insensitive to the velocity component parallel to the long axis. This is known as the independence principle and holds quite accurately for real attached viscous flows up to angles of yaw between the flow and the long axis from normal flow (0° of yaw) to about 45° of yaw. This covers the usual range for such elements as wind turbine blades. For larger yaw angles than this the independence principle is increasingly in error, and as the yaw angle approaches 90° the flow becomes more like that of a slender body.