A new pressure formulation for splitting methods is developed that results in high-order time-accurate schemes for the solution of the incompressible Navier-Stokes equations. In particular, improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods. A new family of stiffly stable schemes is employed in mixed explicit/implicit time-intgration rules. These schemes exhibit much broader stability regions as compared to Adams-family schemes, typically used in splitting methods. Their stability properties remain almost constant as the accuracy of the integration increases, so that robust third- or higher-order time-accurate schemes can readily be constructed that remain stable at relatively large CFL number. The new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation. A model Stokes problem is studied in detail, and several examples of Navier-Stokes solutions of flows in complex geometries are reported. Comparison is made with the previously used first-order in time spectral element splitting and non-splitting (e.g., Uzawa) schemes. High-order splitting/spectral element methods combine accuracy in space and time, and flexibility in geometry, and thus can be very efficient in direct simulations of turbulent flows in complex geometries.
Bibliographical noteFunding Information:
We acknowledge the contribution of A. Tomboulides in implementing some of the computer codes for this work. Financial support for the current work was provided by grants from NSF (CTS-8906911 and CTS-8906432) DARPA Contract NOOO14-86-K-0759, and ONR Contracts NOOOl4-82-C-0451 and NOO14-90-1315. One of us (M.I.) acknowledges the support of the Fund for Promotion of Research at the Technion.