Michael Graham - Wind Energy Handbook

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(10 5) to O(10 8)] 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.

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 body constricting the streamlines and hence increasing flow speed. Part in the case of a body generating lift is due to the fall in pressure associated with the lift or circulation described above. The increase in flow speed on the ‘upper’ or ‘suction’ surface when the body is an aerofoil section at a significant angle of attack to the ambient flow is much greater than on the ‘lower’ or ‘pressure’ surface. Following the suction and velocity peak, the flow on the upper surface must slow down again to reach near‐ambient pressure conditions before streaming off into the wake. As the flow slows the pressure rises, and this ‘adverse’ streamwise pressure gradient acting on the much reduced momentum in the flow layers very close to the surface within the boundary layers further reduces their momentum, eventually to zero and if strong enough to a reverse flow, although the external flow may not yet have even slowed to ambient; see Figure A3.3. The process is opposed by viscous mixing with higher momentum from the external flow. But if the adverse pressure gradient is strong enough, reversed flow occurs in the boundary layer. This is known as separation and the boundary layer separates from the surface at that point. The separated region becomes much thicker and dramatically alters the pressure distribution around the body. This strongly affects both the lift force, even causing it to fall abruptly, and the near balance of the front and rear streamwise components of the integrated pressures, causing the pressure drag to increase rapidly to much larger values. The phenomenon is known as the stall condition for the aerofoil. A boundary layer that does not separate from the surface before it reaches the downstream end of the surface (the trailing edge on an aerofoil) is termed unseparated . It finally sheds (or separates) from this downstream edge by virtue of the sudden change of surface slope. Flow around any sharp edge is not sustainable, because this would generate a very high velocity at the edge followed by an extreme adverse pressure gradient as the flow slows down again. In the case of streamlined (i.e. unseparated) flow over an aerofoil, both surface boundary layers remain unseparated until they meet at the trailing edge, from which they convect together downstream in a thin wake, and the pressure drag remains very small.

Figure A3.3 Separation of a boundary layer.

Figure A3.4 Separated flow past a flat plate.

On some bluff (i.e. non‐streamlined) bodies, the boundary layers separate from different downstream edges and do not meet up, such as is shown in Figure A3.4for a flat plate normal to the flow. In these cases, as for the cases of boundary layer separation from continuous surfaces, a thick wake results, which often contains large eddying motions, and the pressure drag is high. A sharp edge on a body will always cause separation. For the flat plate broad‐side onto the flow, Figure A3.4, the boundary layer separates at the sharp edges and C Dis almost independent of Re but is dependent upon the plate's aspect ratio.