A SECOND-ORDER NEWTON-RAPHSON METHOD FOR IMPROVED NUMERICAL STABILITY IN THE DETERMINATION OF DROPLET SIZE DISTRIBUTIONS IN SPRAYS

摘要

The maximum entropy principle method has been very popular,and it has achieved reasonable success predicting droplet size and velocity distribution in sprays in the past two decades.The recently proposed method,maximization of entropy generation,takes into account the irreversibility during the atomization process,and is more consistent with the physics involved.Both of these methods generate models consisting of implicit,highly nonlinear equations involved with exponential functions and integrals.The classical Newton's method has traditionally been adopted as the solver;however,its inherent disadvantage is the requirement that the initial guess for the successive iteration in the numerical solution process be sufficiently close to the solution,otherwise the iteration may diverge rapidly.This study introduces a modification to the classical Newton's method with the Newton's second-order method and the successive under-relaxation(SUR)technique.Three other algorithms based on the Newton's method are also compared with the above methods.Results show that the proposed second-order Newton's method and the SUR technique can greatly improve the numerical stability and,indeed,relinquish the strict requirement on the initial guess.