Burrill’s procedure is essentially a strip theory method of analysis which combines the basic principles of the momentum and blade element theories with certain aspects of the vortex analysis method. As such the method works quite well for moderately loaded pro- pellers working at or near their design condition;
Theoretical methods – propeller theories 175 however, for heavily and lightly loaded propellers the
correlation with experimental results is not so good, although over the years several attempts have been made to improve its performance in these areas; for example, Sontvedt (Reference 12).
In developing his theory, Burrill considered the flow through an annulus of the propeller, as shown in Figure 8.5. From considerations of continuity through the three stations identified (far upstream, propeller disc and far downstream) one can write, for the flow through the annulus in one revolution of the propeller,
2πr0
Va
n δr0=2πr1Va(1+Kβia) n δr1
=2πr2
Va(1+2Kεa) n δr2
whereKβi andKεare the Goldstein factors at the pro- peller disc and in the ultimate wake respectively. Also Vais the speed of advance of the uniform stream anda is the axial induction factor.
From this relation, by assuming that the inflow is con- stant at each radius, that isa=constant, it can be shown that the relationship between the radii of the slipstream is given by
r0=r1(1+Kβia)1/2=r2(1+Kεa)1/2 (8.18) Furthermore, it is also possible, by considering the momentum of the fluid in relation to the quantity flow- ing through the annular region, to define a relation for the thrust acting on the fluid.
dT =VdQ
for which the thrust at the propeller disc can be shown to be
dT =4πr1KεaVA2[1+Kβia]dr1 (8.19) However, by appealing to the blade element concept, an alternative expression for the elemental thrust at a particular radiusr1can be derived as follows:
dT=ρ
2Zc(1+a)2
× VA2 sin2βi
[c1cosβi−cdsinβi] dr1 (8.20) and then by equating these two expressions, equations (8.19) and (8.20), and in a similar manner deriving two further expressions for the elemental torque acting at the particular radius, one can derive the following pair of expressions for the axial for tangential inflow factors aandarespectively.
a 1+a
1+Kβia 1+a
= c1σs
2Kε
cos(βi+γ) 2sin2βicosγ a
1−a
1−Kβia 1+a
= c1σs
2Kε
sin(βi+γ) sin 2βicosγ
⎫⎪
⎪⎬
⎪⎪
⎭ (8.21)
in whichσs is the cascade solidity factor defined by Zc/(2πr) andγis the ratio of drag to lift coefficientcd/c1. In equations (8.21) the lift coefficientc1is estimated from the empirical relationships derived from wind tun- nel tests on aerofoil sections and applied to the results of thin aerofoil theory as discussed in the preceding chapter. Thus the lift coefficient is given by
cl=2πks·kgs(α+α0) (8.22) whereksandkgsare the thin aerofoil to single aerofoil and single aerofoil to cascade correction factors for lift slope derived from wind tunnel test results andα0 is the experimental zero lift angle shown in Figure 8.5(b).
The termαis the geometric angle of attack relative to the nose–tail line of the section as seen in Figure 8.5(c).
Form equation (8.22) it can be seen that the lift slope curve reduces from 2π in the theoretical thin aerofoil case to 2πkskgs in the experimental cascade situation;
that is, the lineCCin Figure 8.5(b). Burrill chooses to expressks as a simple function of thickness to chord ratio, andkgsas a function of hydrodynamic pitch angle βiand cascade solidityσs. Subsequently work by van Oossanen, discussed in Chapter 7, has shown that these parameters, in particularksandkgs, are more reliably expressed as functions of the section boundary layer thicknesses at the trailing edge of the sections.
With regard to the effective angle of attack of the section, this is the angle represented by the line CD measured along the abscissa of Figure 8.5(b) and the angle (α+α0) on Figure 8.5(c). This angleαeis again calculated by appeal to empirically derived coefficients as follows:
αe=α+α0=α+α0TH(Kα0−Kgα
0) (8.23) whereα0THis the two-dimensional theoretical zero lift angle derived from equation (7.27) andKα0 andKgα0 are the single aerofoil to theoretical zero lift angle cor- relation factor and cascade allowance respectively. The former, as expressed by Burrill, is a function of thick- ness to chord ratio and position of maximum camber whilst the latter is function of hydrodynamic pitch angle βiand solidity of the cascadeσs.
Now by combining equations (8.21) and (8.22) and noting that the term (1+Kβia)/(1 +a) can be expressed as
1−[(1−Kβi)tan(βi−β)/tanβi]
it can be shown that the effective angle of attack (α+α0) is given by
α+α0= 2 KsKgsπσs
Kεsinβitan(βi−β)
× 1−tan(βi−β) tanβi
(1−Kβi)
(8.24) This equation enables, by assuming an initial value, the value of the effective angle of attack to be calculated for any given value of (βi−β), by means of an iterative
Ch08-H8150.tex 16/5/2007 18: 52 page 176
176 Marine propellers and propulsion
Figure 8.5 The Burrill analysis procedure: (a) slipstream contraction model; (b) lift evaluation and (c) flow vectors and angles
Theoretical methods – propeller theories 177 process. Once convergence has been achieved the lift
can be calculated from equation (8.22). The drag coef- ficient again is estimated from empirical data based on wind tunnel test results and this permits the calculation of the elemental thrust and torque loading coefficients:
dKQ= π3x4σs
8 (1−a)2(1−tan2βi)c1
sin(βi+γ) cosγ dx dKT=
dKQ dx
2 xtan(βi+γ)dx
⎫⎪
⎪⎬
⎪⎪
⎭ (8.25) where the rotational induced velocity coefficientacan most conveniently be calculated from
a=(tanβi−tanβ) tan(βi+γ)
1+tanβitan(βi+γ) (8.26) Figure 8.6 shows the algorithm adopted by Burrill to calculate the radial distribution of loading on the pro- peller blade, together with certain modifications, such as the incorporation of the drag coefficients from his later paper on propeller design (Reference 13).
Burrill’s analysis procedure represents the first coher- ent step in establishing a propeller calculation proced- ure. It works quite well for the moderately loaded propeller working at or near its design condition; how- ever, at either low or high advance the procedure does not behave as well. In the low advance ratios the con- stant radial axial inflow factor, consistent with a lightly loaded propeller, must contribute to the underpredic- tion of thrust and torque coefficient for these advance ratios. Furthermore, the Goldstein factors rely on the conditions of constant hydrodynamic pitch and conse- quently any significant slipstream distortion must affect the validity of applying these factors. Alternatively, in the lightly loaded case it is known that the theory breaks down when the propeller conditions tend toward the production of ring vortices.
Clearly since the Goldstein factors are based on the concept of zero hub radius, the theory will benefit from the use of the Tachmindiji factors, which incorporate a boss of radius 0.167R. Additionally, the use of the particular cascade corrections used in the method were criticized at the time of the method’s publication; how- ever, no real alternative for use with a method of this type has presented itself to this day.
The Burrill method represents the final stage in the development of a combined momentum–blade element approach to propeller theory. Methods published sub- sequent to this generally made greater use of the lifting line, and subsequently lifting surface concepts of aero- dynamic theory. The first of these was perhaps due to Hill (Reference 14) and was followed by Strscheletsky (Reference 15); however, the next significant devel- opment was that due to Lerbs (Reference 16), who laid the basis for moderately loaded lifting line theory.
Strscheletsky’s work, although not generally accepted at
the time of its introduction, due to the numerical com- plexity of his solution, has subsequently formed a basis for lifting line heavily loaded propeller analysis.