Airscrew coefficients

The performance of an airscrew may be determined by model tests. As is the case with all model tests it is necessary to find some way of relating these to the full-scale performance, and dimensional analysis is used for this purpose. This leads to a number of coefficients, analogous to the lift and drag coefficients of a body. These coefficients also serve as a very convenient way of presenting airscrew performance data, which may be calculated by blade-element theory (Section 9.4), for use in aircraft design.

9.2.1 Thrust coefficient

Consider an airscrew of diameter D revolving at n revolutions per second, driven by a torque Q, and giving a thrust of T. The characteristics of the fluid are defined by its density, p, its kinematic viscosity, v, and its modulus of bulk elasticity, K. The forward speed of the airscrew is V. It is then assumed that

T = h{D, n,p, v, K, V)

= CDanbpcvdKeVf (9.15)

Then, putting this in dimensional form,

[MLT~2] = [(L)fl(T)_*(ML_3)c(L2T~1)‘/(ML_IT_2)‘?(LT-1/’]

Separating this into the three fundamental equations gives

(M) 1 =c + e

(L) 1 = a-3c + 2d-e+f

(T) 2 = b + d + 2e+f

Solving these three equations for a, b

a = 4-2e-2d-f b = 2-d-2e-f c = 1-е

Substituting these in Eqn (9.15) gives

(9.16)

Consider the three factors within the square brackets.

(i) v/D[76] n; the product Dn is a multiple of the rotational component of the blade tip speed, and thus the complete factor is of the form v/(length x velocity), and is therefore of the form of the reciprocal of a Reynolds number. Thus ensuring equality of Reynolds numbers as between model and full scale will take care of this term.

(ii) KjpLP-n2-, K/p = a2, where a is the speed of sound in the fluid. As noted above, Dn is related to the blade tipspeed and therefore the complete factor is related to (speed of sound/velocity)2, i. e. it is related to the tip Mach number. Therefore care in matching the tip Mach number in model test and full-scale flight will allow for this factor.

(iii) V/nD; V is the forward speed of the airscrew, and therefore V/n is the distance advanced per revolution. Then VjnD is this advance per revolution expressed as a multiple of the airscrew diameter, and is known as the advance ratio, denoted by J.

Thus Eqn (9.16) may be written as

T = Cpn2D4h(Re, M, J) (9.17)

The constant C and the function h(Re, M, J) are usually collected together, and denoted by кт, the thrust coefficient. Thus, finally.

T = kTpn2D4 (9.18)

кт being a dimensionless quantity dependent on the airscrew design, and on Re, M and J. This dependence may be found experimentally, or by the blade-element theory.