Michael Graham - Wind Energy Handbook

The circulatory component of this flow is shown in Figure A3.8and has the same distribution of velocity outside the boundary layer as a line vortex.

The lift force due to circulation is given by the Kutta–Joukowski theorem, called after the two pioneering aerodynamicists who, independently, realised that this was the key to the understanding of the phenomenon of lift generated in subsonic flow on all bodies, including the spinning cylinder:

(A3.1)

Figure A3.7 Flow past a rotating cylinder.

Figure A3.8 Circulatory flow round a rotating cylinder.

Here Γ is the circulation, or vortex strength, defined as the integral

(A3.2)

around any path enclosing the body, and v is the velocity tangential to the path s .

Two‐dimensional inviscid potential flow about a general 2‐D body section is non‐unique and is only fixed by defining where the flow separates. A non‐rotating body can have a circulatory flow about any section, the circulation being controlled by where on the section the boundary layers separate. On an aerofoil section, pre‐stall, the sharp trailing edge is the only edge at which the flow separates. Such a flow about an aerofoil as shown in Figure A3.9can be composed of (i) a non‐circulatory flow induced by the approaching free stream, and (ii) a purely circulatory flow that is equivalent to a distribution of vorticity around the section. In general, neither of these flows separate from the trailing edge, i.e. appropriately, but by adding a suitable amount of the latter, thus fixing the circulation, to the former (iii) a composite flow is obtained that does separate from the trailing edge. The condition enforcing separation of the inviscid flow from the trailing edge is known as the Kutta–Joukowski condition . At large distances radially from the axis of a (quasi‐) 2‐D body, the flow field is a combination of the uniform incident flow with a vortex flow if the body has lift (and a line‐source flow if it has significant viscous drag.) The v ‐component of the free stream U in Eq. (A3.2)(and similarly the source flow component if present) integrates around the closed circuit to zero. The v ‐component due to a line vortex, taken, for example, on a circular path concentric with the vortex, is v = k/r, where k is a constant. This integrates around the circuit in Eq. (A3.2)to the circulation:

Figure A3.9 Flow past an aerofoil at a small angle of attack: (a) inviscid flow, (b) circulatory flow, and (c) real flow.

(easily seen for circular circuits defined by constant r, but true for all circuits enclosing the vortex). Hence the section lift/unit span is

In the case of streamlined lifting bodies such as aerofoils, the circulation Γ that is fixed by the Kutta–Joukowski condition at the trailing edge can be shown to increase with angle of attack α in proportion to sin α. Although the velocities and pressures above and below the aerofoil at the trailing edge must be the same, the particles that meet there are not the same ones that parted company at the leading edge. The particle that travelled over the aerofoil upper surface, even though a longer distance, normally reaches the trailing edge before the one travelling over the shorter lower surface because its speed‐up by the circulation is proportionately greater.

In the corresponding case of a real viscous flow, the boundary layers separate at the trailing edge as discussed earlier, very closely approximating this condition. Thus, pre‐stall lift on an aerofoil section in real flow is quite accurately predicted by inviscid potential flow analysis. However, inviscid flow analysis does not predict the drag, the inviscid (profile) drag being identically zero because in this case the section of itself generates no wake.

The pressure variation (minus the ambient static pressure of the undisturbed flow) around an aerofoil is shown in Figure A3.10. The upper surface is subject to suction (with the ambient pressure subtracted) and is responsible for most of the lift force. The pressure distribution is calculated without the presence of the boundary layer because the normal pressure difference across the boundary layer is small enough to be neglected. Higher order, more accurate solutions for the pressures and forces may be obtained by taking account of the effect of the slowed velocity in the boundary layer displacing the streamlines of the quasi‐inviscid flow outwards by a small amount like a small thickness addition to the profile.

Figure A3.10 The pressure distribution around the NACA0012 aerofoil at α = 5° (shown schematically around the aerofoil).

Figure A3.11shows the same distribution with the pressure coefficient ( ) plotted against the chordwise coordinate of the aerofoil profile: the full line shows the pressure distribution if the effects of the boundary layer are ignored, and the dashed line shows the actual distribution.

Figure A3.11 The pressure distribution around the NACA0012 aerofoil at α =5° (pressure coefficient C Pvs x/c).

The effect of the boundary layer is to modify the pressure distribution at the rear of the aerofoil such that lower pressure occurs there than if there is no boundary layer. There is no stagnation pressure at the trailing edge, where the pressure tends to be much closer to ambient. The boundary‐layer‐modified pressure distribution gives rise to pressure drag that is added to the skin friction drag, also caused by the boundary layer.

If the angle of attack exceeds a certain critical value (typically 10° to 16°, depending on the Re ), separation of the boundary layer on the ‘suction’ (or upper) surface takes place. A wake forms above the aerofoil starting from this separation ( Figure A3.12), and the circulation and hence the lift are reduced and the drag increased. The flow past the aerofoil has then stalled. A flat plate at an angle of attack will also behave like an aerofoil and develop circulation and lift but will stall at a very low angle of attack because of the sharp leading edge. Cambering (or curving) the plate will increase the angle of attack for stall onset, but a much greater improvement can be obtained by giving thickness to the aerofoil together with a suitably rounded leading edge.