  We propose a novel collocated projection method for solving the
incompressible Navier-Stokes equations with arbitrary boundaries. Our approach
employs non-graded octree grids, where all variables are stored at the nodes.
To discretize the viscosity and projection steps, we utilize supra-convergent
finite difference approximations with sharp boundary treatments. We demonstrate
the stability of our projection on uniform grids, identify a sufficient
stability condition on adaptive grids, and validate these findings numerically.
We further demonstrate the accuracy and capabilities of our solver with several
canonical two- and three-dimensional simulations of incompressible fluid flows.
Overall, our method is second-order accurate, allows for dynamic grid
adaptivity with arbitrary geometries, and reduces the overhead in code
development through data collocation.
