#### Vol. 10, No. 1, 2015

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An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries

### David Trebotich and Daniel T. Graves

Vol. 10 (2015), No. 1, 43–82
##### Abstract

We present an adaptive, finite volume algorithm to solve the incompressible Navier–Stokes equations in complex geometries. The algorithm is based on the embedded boundary method, in which finite volume approximations are used to discretize the solution in cut cells that result from intersecting the irregular boundary with a structured Cartesian grid. This approach is conservative and reduces to a standard finite difference method in grid cells away from the boundary. We solve the incompressible flow equations using a predictor-corrector formulation. Hyperbolic advection terms are obtained by higher-order upwinding without the use of extrapolated data in covered cells. The small-cell stability problem associated with explicit embedded boundary methods for hyperbolic systems is avoided by the use of a volume-weighted scheme in the advection step and is consistent with construction of the right-hand side of the elliptic solvers. The Helmholtz equations resulting from viscous source terms are advanced in time by the Crank–Nicolson method, which reduces solver runtime compared to other second-order time integrators by a half. Incompressibility is enforced by a second-order approximate projection method that makes use of a new conservative cell-centered gradient in cut cells that is consistent with the volume-weighted scheme. The algorithm is also capable of block structured adaptive mesh refinement to increase spatial resolution dynamically in regions of interest. The resulting overall method is second-order accurate for sufficiently smooth problems. In addition, the algorithm is implemented in a high-performance computing framework and can perform structured-grid fluid dynamics calculations at unprecedented scale and resolution, up to 262,144 processor cores. We demonstrate robustness and performance of the algorithm by simulating incompressible flow for a wide range of Reynolds numbers in two and three dimensions: Stokes and low Reynolds number flows in both constructed and image data geometries ($Re\ll 1$ to $Re=1$), flow past a cylinder ($Re=300$), flow past a sphere ($Re=600$) and turbulent flow in a contraction ($Re=6300$).

##### Keywords
incompressible Navier–Stokes, embedded boundary method, finite volume method, cut cell method, projection method, adaptive mesh refinement
##### Mathematical Subject Classification 2010
Primary: 35K57, 35Q35, 76D05, 76D07