SBIR Phase I: A Unified Coordinates Approach to Gridless Computation

This Small Business Innovation Research (SBIR) Phase I research project will develop an innovative unified coordinates approach to gridless computation. Instead of using the spatially fixed Eulerian coordinate system as in most Computional Fluid Dynamics (CFD) computation, the proposed unified coordinates approach is based on a generalized coordinate system which moves with velocity hq with q as the velocity of fluid particles and h as a free function. It includes the Eulerian coordinates approach as a special case when h=0 and the Lagrangian when h=1. Choosing h to preserve grid orthogonality can result in a coordinate system which avoids not only excessive numerical diffusion across slip lines as in the Eulerian coordinates but also severe grid deformation as in the Lagrangian coordinates. Unlike Arbitrary Lagrangian-Eulerian (ALE) approach, no remapping from the distorted Lagrangian grid onto the spatially fixed Eulerian grid is required. All computations are done entirely in the transformed space without a staggered grid. More importantly, using the unified coordinates, the computational grid is generated simultaneously by the flow (more precisely by the movement of the pseudo particles) while computing the flow field. Therefore, the computational grid is no longer a required input for CFD computation and gridless CFD computation becomes feasible. The traditional CFD methods are based on the spatially fixed Eulerian coordinates approach, in which body-fitted computational grids are needed a priori to accurately implement surface boundary conditions. The generation of body-fitted grids around complicated geometries in real applications is a very tedious process, which requires substantial human intervention and hence experience and specialized training, making CFD as somewhat of an art. The proposed SBIR effort will resolve this problem and significantly facilitate the use of CFD in manufacturing industry as a design tool.