Question 8 of 9: Wake-Field Effects on Blade Section Loading

Question 8: Wake-Field Effects on Blade Section Loading (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Wake-fraction iso-lines at the $0.7R$ blade path (starboard side, $\theta=0$ at top-dead-centre through $\theta=\pi$ at bottom-dead-centre, in steps of $\pi/4$), read from the exam's wake-field sketch: point A ($\theta=0$) near $w\approx0.80$; point B ($\theta=\pi/4$) near $w\approx0.60$; point C ($\theta=\pi/2$) near $w\approx0.80$; point D ($\theta=3\pi/4$) near $w\approx0.40$; point E ($\theta=\pi$) near $w\approx0.10$. The blade section rotates at constant angular speed, so its own tangential (rotational) speed $U=2\pi nr$ at $0.7R$ is constant; only the axial inflow $V_a(r,\theta)=V_S(1-w(r,\theta))$ varies with position.

Find. The resultant-velocity/angle-of-attack picture (i), the resulting lift-coefficient variation (ii), and the resulting elemental thrust-load variation over one revolution (iii), for the blade section at $0.7R$.

Approach. At each angular position the blade section sees a resultant velocity that is the vector sum of the (constant) rotational speed $U$ and the (position-dependent) axial inflow $V_a(\theta)$; a lower $V_a$ steepens the resultant vector relative to the blade's fixed pitch line, raising the angle of attack, which (below stall) raises the lift coefficient and hence the elemental thrust — so the wake pattern is mapped, position by position, through to a cyclic thrust load.

(i) Resultant velocity and angle of attack. Because $U$ is fixed while $V_a$ varies with $\theta$, the local inflow angle $\beta=\tan^{-1}(V_a/U)$ is smallest where $w$ is largest (A and C, both $w\approx0.80$) and largest where $w$ is smallest (E, $w\approx0.10$). Since the blade's geometric pitch angle $\varphi$ at this radius is fixed, the angle of attack $\alpha=\varphi-\beta$ moves opposite to $\beta$: $\alpha$ is highest at A and C (low $V_a$, low $\beta$) and lowest at E (high $V_a$, high $\beta$), with B and D at intermediate values consistent with their intermediate wake fractions.

(ii) Lift coefficient. Below stall, $C_L$ rises with $\alpha$; the blade section therefore experiences its highest $C_L$ passing through the high-wake top region (A, C) and its lowest $C_L$ passing through the bottom (E), with B and D on the rising/falling flank.

(iii) Elemental thrust load over one revolution. Since elemental thrust at a blade section is governed primarily by lift, the same once-per-revolution pattern carries through directly to loading: the blade element is most heavily loaded passing through the top of the disk (A, C — high wake, high $\alpha$, high $C_L$) and least loaded passing the bottom (E — low wake, low $\alpha$, low $C_L$), with B and D marking the transition. This cyclic loading — once per revolution for the mean pattern, and at blade-passing frequency ($Z\times$ shaft speed) for the combined effect of all four blades — is the direct source of propeller-induced hull vibration and unsteady bearing/shaft forces, and (tying back to Question 6) the local loading peak near top-dead-centre is exactly where transient cavitation is most likely to be triggered even in a design whose time-averaged Burrill check is satisfactory.