Application to a Hovering Rotor

These general equations of fluid mass, momentum, and energy conservation may now be applied to the specific problem of a hovering helicopter rotor. The procedures are basically those attributed to Rankine and Froude, as generalized by Glauert (1935) and adopted by others, including Johnson (1980). In Fig. 2.5 let cross section 0 denote the plane far upstream of the rotor, where in the hovering case the fluid is quiescent (i. e., Vo = 0). The rotor disk area is denoted by A. Cross sections 1 and 2 are the planes just above and below the rotor disk, respectively, and the “far” wake[11] is denoted by cross section oo. At the plane of the rotor, assume that the velocity (the induced velocity or velocity imparted to the mass of air contained in the control volume at the rotor disk) is vt. In the far wake

(the vena contracta), the velocity will be increased over that at the plane of the rotor and this velocity is denoted by w.

From the assumption that the flow is quasi-steady and by the principle of conservation of mass, the mass flow rate, m, must be constant within the boundaries of the rotor wake (control volume). Therefore, the mass flow rate is

rh = /f (2-4)

and the 1-D incompressible flow assumption reduces this equation to

m = pAooW = pA2Vi = pAvi. (2.5)

The principle of conservation of fluid momentum gives the relationship between the rotor thrust, T, and the net time rate-of-change of fluid momentum out of the control volume (Newton’s second law). The rotor thrust is equal and opposite to the force on the fluid, which is given by

-F = T = ff p{V ■ dS)V – ff p{V ■ dS)V. (2.6)

J J oo J Jo

Because in hovering flight the velocity well upstream of the rotor is quiescent, the second term on the right-hand side of the above equation is zero. Therefore, for the hover problem, the rotor thrust can be written as the scalar equation

From the principle of conservation of energy, the work done on the rotor is equal to the gain in energy of the fluid per unit time. The work done per unit time, or the power consumed by the rotor, is T u, and this results in the equation

(2.8)

In hover, the second term on the right-hand side of the above equation is zero so that

Tvi = If – p(V ■ dS)Vz = – raw2. J J oo 7 2

From Eqs. 2.7 and 2.9 it is clear that

(2.Ю)

or that w = 2Vi. This, therefore, gives a simple relationship between the induced velocity in the plane of the rotor, u, , and the velocity w in the vena contracta.

Rotor Slipstream

Because the flow velocity increases in the wake below the rotor, continuity con­siderations require that the area of the slipstream must decrease. This is apparent from the empirical observations in Figs. 2.2 and 2.4. It follows from the conservation of fluid mass between the rotor and the vena contracta that

pAvi = pAooW = pA^ilvi) = IpAooVi,

so that in hover the ratio of the cross-sectional area of the fully developed far wake to the area of the rotor disk is

-^OO 1

~A ~2

In other words, based on ideal fluid flow assumptions, the vena contracta is an area that is exactly half of the rotor disk area. Alternatively, by considering the radius of the far rotor wake, Гоо, relative to that of the rotor, R, it is easy to show from mass conservation considerations that

Therefore, the ratio of the radius of the wake to the radius of the rotor is 1 /V2 = 0.707. This is called the wake contraction ratio. In practice, it has been found experimentally that the wake contraction ratio is not as much as the theoretical value given by the momentum theory; typically it is only about 0.78 compared to 0.707. This is mainly a consequence of the viscosity of the fluid, the reality that a nonuniform inflow will be produced over the disk and a small swirl component of velocity in the rotor wake induced by the spinning rotor blades. Behaviors directly attributable to the viscosity of the fluid are termed nonideal effects, and these will be considered in detail later.