We have developed an explicit and direct pressure-correction method for simulating incompressible flows over curved walls. In order to integrate the Navier Stokes (NS) equations in time, we have developed an explicit, three-stage, third-order Runge-Kutta based projection-method (FastRK3) which requires solving the Poisson equation for pressure only once per time step. We have chosen to discretize the incompressible NS equations written in the orthogonal coordinates rather than the general formulation in curvilinear coordinates because the former does not contain cross-derivatives in the advection, diffusion, Laplacian, and gradient operators. Thus, the computational cost of solving the NS equations is substantially reduced and the numerical stencils of the finite difference approximations to these operators mirror that of the Cartesian formulation. This property also allows us to develop an FFT-based Poisson solver for pressure (FastPoc) for those cases where the components of the metric tensor are independent of one spatial direction: surfaces of linear translation (e.g., curved ramps and bumps) and surfaces of revolution (e.g., axisymmetric ramps). We have verified and validated FastRK3 and we have applied FastRK3 for simulating separated flows over ramps and a bump. Finally, our results show that the new FFT-based Poisson solver, FastPoc, is thirty to sixty times faster than the multigrid-based linear solver (depending on the tolerance set for the multigrid solver), and the new flow solver, FastRK3, is overall four to seven times faster when using FastPoc rather than multigrid. In summary, given that the computational mesh satisfies the property of orthogonality, FastRK3 can simulate flows over curved walls with second-order accuracy in space.

我们已经开发了一种显式且直接的压力校正方法，用于模拟弯曲壁上不可压缩的流动。为了及时整合Navier Stokes（NS）方程，我们开发了一种显式的基于三阶三阶Runge-Kutta的投影方法（FastRK3），该方法仅需在每个时间步上求解一次泊松方程的压力即可。 。我们选择离散化以正交坐标而不是曲线坐标形式表示的不可压缩NS方程，因为前者在对流，扩散，拉普拉斯算子和梯度算子中不包含交叉导数。因此，大大减少了求解NS方程的计算成本，并且这些算子的有限差分近似的数值模版反映了笛卡尔公式。此特性还允许我们针对度量张量的分量与一个空间方向无关的情况开发基于FFT的泊松解压器（FastPoc）：线性平移表面（例如，弯曲的斜坡和凸点）和旋转（例如，轴对称的斜坡）。我们已经验证并验证了FastRK3，并已应用FastRK3来模拟斜坡和凹凸上的分离流。最后，我们的结果表明，新的基于FFT的泊松求解器FastPoc比基于多网格的线性求解器快30到60倍（取决于多网格求解器的容差设置），而新的流量求解器FastRK3是使用FastPoc而不是多重网格时，总体速度要快四到七倍。总而言之，鉴于计算网格满足正交性，