Downwash

Let us consider the aerofoil with hypothetical spanwise variation of circulation due to the combined bound vortex filaments as shown in Figure 8.4. At some point y along the span, an induced velocity equal to:

Подпись: Swy1 = —

f ‘(y)dy 4n(y — yO ‘

will be felt in the downward direction. All elements shed vorticity along the span and add their contribution to the induced velocity at y1 so that the total influence of the trailing system at y1 is:

_ 1 rb ГШу

Wyl 4n J—b (y — y1) ’

Figure 8.3 Spanwise distribution of bound vortex filaments.

 

Figure 8.4 Spanwise variation of the strength of the combined bound vortex filaments.

 

Sk

 

w

Figure 8.5 Variation of downwash caused by the vortex system around an aerofoil.

Подпись: 1 b (dk/dy) dy Wyl 4n .J-b (y — У1) Подпись: (8.2)

that is:

The induced velocity at y1, in general, is in the downward direction and is called downwash.

The downwash has the following two important consequences which modify the flow about the aerofoil and alter its aerodynamic characteristics:

• The downwash at y1 is felt to a lesser extent ahead of y1 and to a greater extent behind, and has the effect of tilting the resultant wind at the aerofoil through an angle:

Подпись:(8.3)

The downwash around an aerofoil will be as illustrated in Figure 8.5.

The downwash reduces the effective incidence so that for the same lift as the equivalent infinite or two-dimensional aerofoil at incidence a, an incidence of a = a^ + e is required at that section of the aerofoil. Variation of downwash in front of and behind an aerofoil will be as shown in Figure 8.5. As illustrated in Figure 8.5, the downwash will diminish to zero at locations far away from the leading edge and will become almost twice of its magnitude wcp at the center of pressure, downstream of the trailing edge.

• In addition to this motion of the air stream, a finite aerofoil spins the air flow near the tips into what eventually becomes two trailing vortices of considerable core size. The generation of these vortices requires a quantity of kinetic energy. This constant expenditure of energy appears to the aerofoil as the trailing vortex drag.

Figure 8.6 shows the two velocity components of the relative wind superimposed on the circulation generated by the aerofoil. In Figure 8.6, LOT is the two-dimensional lift, VR is the resultant velocity and V is the freestream velocity. Note that the two-dimensional lift is normal to VR and the actual lift L is normal to V. The two-dimensional lift is resolved into the aerodynamic forces L and Dv, respectively, normal and against the direction of the velocity V of the aerofoil. Thus, an important consequence of the downwash is the generation of drag Dv. Also, as illustrated in Figure 8.6, the vortex system inducing downwash w tilts the aerofoil in the nose-up direction. In Figure 8.6, V is the forward speed of aerofoil, VR is the resultant velocity at the aerofoil, a is the incidence, e (= w/V) is the downwash angle, aOT = (a — e), the equivalent two-dimensional incidence and Dv is the trailing vortex drag. The trailing vortex drag is also referred to as vortex drag or induced drag.

Подпись: (8.4)

The forward wind velocity generates lift and the downwash generates the vortex drag Dv:

This shows that there is no vortex drag if there is no trailing vorticity.

As a consequence of the trailing vortices, which are produced by the basic lifting action of a (finite span) wing, the wing characteristics are considerably modified, almost always adversely, from those of the equivalent two-dimensional wing of the same section. A wing whose flow system is closer to the two-dimensional case will have better aerodynamic characteristics than the one where the end effects are conspicuous. That is, large aspect ratio aerofoils are better than short span aerofoils.