# Fully Explicit Projection Method for Incompressible flow

## Motivation

본 연구는 비압축성 유체를 수치적으로 해석하기 위한 새로운 수치적 방법에 대한 내용이다. 유동에 대한 지배방정식인 Navier-Stokes equation의 continuity 조건을 만족시키기 위해 압력에 대한 Poisson equation을 풀어야 하는데, 기존의 implicit solver는 많은 계산비용이 요구된다. 이를 해결하기 위해 explicit한 방법을 모색하였다.

## Details

The Poisson equation for pressure arising from nonzero divergence of the nonlinear term in the integration of the Navier-Stokes equations requires a lot of computational cost except for cases with periodic domain. In order to mitigate this cost, we propose a new project algorithm which is fully explicit, thus not requiring iterations. The projection operator, , which projects any vector field with divergence into the divergence-free subspace in the Fourier space, when the distance from the point in question. This allows truncation so that the resulting local distribution of the projection operator, through convolution, can be used to obtain projected nonlinear terms which have relatively small divergence. This `approximate’ projection scheme was then applied to direct numerical simulation of isotropic turbulence to investigate effectiveness and efficiency of the scheme in reducing divergence and correct projection of the nonlinear terms through the statistical properties of the turbulent flow.

#### Test results in isotropic turbulence

Used number of coefficients:
dt=0.001
Maximum iteration number: 3000(1sec)
Record: 600(0.2sec)
`                 Correlation                                                Energy spectrum `