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.
A3.2 The boundary layer
The velocity of the flow adjacent to the surface of any solid body, and in particular wind turbine blades and aerofoils, reduces to zero relative to the body at its surface (the no‐slip condition) due to viscous stresses in the fluid. At usual flow Reynolds numbers [O(105) to O(108)] occurring in practice, diffusion is much slower than streamwise convection. As a result nearly all of the change in velocity takes place in very thin regions next the body surface called boundary layers, which therefore exhibit a strongly sheared velocity profile; see Figure A3.2. These boundary layers grow in thickness from the attachment point and are shed eventually into the wake of the body. They convect downstream as free shear layers, forming a wake where viscous stresses are similarly significant. Outside the boundary layers and wake the flow behaves almost as if inviscid. The integrated streamwise component of the skin friction on the body surface due to the viscous stresses gives rise to an important component of the drag on the body, the skin friction drag. The other component is the pressure drag (the integrated streamwise component of the normal forces on the body surface). This component is small because the front half streamwise component of the pressures on the body nearly balances the downstream half; the thinner the boundary layer, the nearer they are in balance. The pressure drag is usually similar in size to the skin friction drag for streamlined bodies, such as aerofoils, but becomes much larger if boundary layer separation occurs. The combined skin friction and pressure drag for an aerofoil section in 2‐D flow is known as the profile drag. The profile drag coefficient of an aerofoil is quite small for these Reynolds numbers while the flow remains attached, depending weakly on the Reynolds number and the angle of attack.
Figure A3.2 Boundary layer showing the velocity profile.
A3.3 Boundary layer separation
The flow over any body, such as a wing, blade, or aerofoil, that generates lift (conventionally regarded as positive ‘upwards’) does so due to the body geometry causing the streamlines of the flow to curve around it (mainly concave downwards) so that downward momentum is added to the vertical component of the momentum in the flow as it exits the influence of the body. The resulting surface pressure distribution can be understood qualitatively by considering the normal pressure gradient required to balance the flow curvature. Therefore, the pressure must fall from ambient far away from the aerofoil to a lower value on its upper surface and rise from ambient towards the lower surface. Bernoulli's equation for energy [e.g. Eq. (3.5a)] shows that decreasing pressure (energy) in a flow must be balanced by increasing kinetic energy, hence increasing velocity, and vice versa. To conserve mass flow rate, higher flow speeds imply streamlines becoming closer together. The general difference in surface flow speed between the upper and lower surfaces of the aerofoil means that any closed circuit integral of flow speed around the body (termed the circulation) is non‐zero. Circulation proportional to the lift is as shown by the Kutta–Joukowski theorem, Eq. (A3.1). A more detailed discussion of circulation is given in Section A3.6. The ‘tighter’ the streamline curvature, as round the nose of an aerofoil section at high angle of attack, the greater the fall in surface pressure resulting in a strong suction peak in this region.
The flow approaching a body such as a blade section has one incident streamline that ‘attaches’ at the front stagnation point where the flow speed falls to zero. The flow speed along the streamline's either side falls to its lowest value close to the body, and pressure there is highest, before the streamline bifurcates, passing either side of the body. Following such a streamline just outside the boundary layer, the flow then rapidly speeds up as it passes over the body surface, to higher values than in the approach flow. Part of this speed‐up is due to the effect of the thickness of the