Study on the Improvement of the Convective Differencing Scheme for the High-Accuracy and Stable Resolution of the Numerical Solution

수치해의 정확성과 안정성이 보장되는 대류항 미분법 개선에 관한 연구

QUICKER scheme has several attractive properties. However, under highly convective conditions, it produces overshoots and possibly some oscillations on each side of steps in the dependent variable when the flow is convected at an angle oblique to the grid line. Fortunately, it is possible to modify the QUICKER scheme using non-linear and linear functional relationship. Details of the development of polynomial upwinding scheme are given in this paper, where it is seen that this non-linear scheme has also third order accuracy. This polynomial upwinding scheme is used as the basis for the SHARPER and SMARTER schemes. Another revised scheme was developed by partial modification of QUICKER scheme using CDS and UPWIND schemes(QUICKUP). These revised schemes are tested at the well known bench mark flows, Two-Dimensional Pure Convection Flows in Oblique-Step, Lid Driven Cavity Flows and Buoyancy Driven Cavity Flows. For pure convection oblique step flow test problem, QUICKUP, SMARTER and SHARPER schemes remain absolutely monotonic without overshoot and oscillation. QUICKUP scheme is more accurate than any other scheme in their relative accuracy. In high Reynolds number Lid Driven Cavity Flow, SMARTER and SHARPER schemes retain lower computational cost than QUICKER and QUICKUP schemes, but computed velocity values in the revised schemes produced less predicted values than QUICKER scheme which is strongly effected by overshoot and undershoot values. Also, in Buoyancy Driven Cavity Flow, SMARTER, SHARPER and QUICKUP schemes give acceptable results.

본 연구에서는 이 기존의 방법들을 개선하여 비균일격자계에서도 사용할 수 있는 비선형 함수관계를 제시하며, 이 방법과 기존의 해법들을 비교할 때 실제 유동장 에서의 해의 정확도 차이, 전산비용의 경제성등은 어느정도 인지를 밝히고, 또 개선된 해법이 앞서 제시한 좀 더 복잡한 유동장에서도 성공적으로 적용 가능한지의 여부를 판단하는 것을 그 목적으로 한다.