Singular Perturbation Method for Supercavitating Propellers

摘要

The singular perturbation method for the supercavitating propeller had four scaling parameters; (1) span length R, (2) chord length c, (3) blade spacing d and (4) cavity length 1 sub c. The first problem solved here assumed that c/R and 1 sub c/R were of order epsilon but d/R was of order of unity. The nature of the singular perturbation problem for such a case was similar to that for the subcavitating propeller solved by Brockett except for the solution of the inner region. The thrust and torque coefficients were obtained explicitly without solving the integral equations. Since the nonlinear supercavitating flow theory was employed in the present work as the inner solution, there existed no limitation for the flow incidence angles or blade profile shapes and thus the present solution would provide more accurate results than those with the linearized theory. The second problem treated here was the case in which c/R was of order of epsilon but 1 sub c/R and d/R were of order of unity. This was the case having long cavities behind the propeller blades so that even when the chord shrank to a line, the cavities were left behind the lifting lines. This portion of cavity sheets was called 'source sheets', the singularity strengths of which were obtained through the cavity sheet matching, a totally different matching procedure from the regular matching. The first-order inner solution used a closure condition, i.e., the total source term S(O) equal to zero.