We derive boundary conditions for the nonlinear incompressible Navier-Stokes equations following the general recipie given in [1]. We present two formulations stemming from different techniques to diagonalize the boundary terms. Both formulations lead to an energy estimate.

In the ﬁrst formulation, the boundary conditions are obtained through a suitable set of rotations. In the second formulation, the boundary conditions are derived directly by a standard eigenvalue decomposition [2, 3]. The two formulations differ in character and have different pro’s and con’s.

The rotational technique lead to more natural formulations, but the formulation must be changed depending on whether there is inﬂow or outﬂow. The rotation formulation does not always lead to a nonlinear bound.

The characteristic technique leads to more involved formulations, but the formulation retains the same form independent of whether there is inﬂow or outﬂow. The characteristic formulation provides a nonlinear bound for the velocity ﬁeld for both solid wall and far ﬁeld boundary conditions.

The continuous problem is approximated by using ﬁnite differences on Summation-By-Parts (SBP) form. The solid wall boundary conditions are weakly imposed with the Simultaneous-Approximation-Term (SAT) procedure [4]. It is shown that by mimicking the continuous analysis, the resulting nonlinear SBP-SAT scheme is provably energy stable, divergence free and high-order accurate.

[1] J. Nordström, “A Roadmap to Well Posed and Stable Problems in Computational Physics,” Accepted in *Journal of Scientiﬁc Computing*.

[2] J. Nordström, N. Nordin, and D. Henningson, “The Fringe Region Technique and the Fourier-method Used in the Direct Numerical Simulation of Spatially Evolving Viscous Flows,” *SIAM Journal of Scientiﬁc Computing*, Vol. 20, No. 4, pp.1365-1393, 1999.

[3] J. Nordström, K. Mattsson, and C. Swanson, “Boundary Conditions for a Divergence Free Velocity-Pressure Formulation of the NavierStokes Equations*,*”* Journal of Computational Physics*, Vol. 225, Issue 1, pp. 874-8901, 2007.

[4] M. Svärd and J. Nordström, “Review of summation-by-parts schemes for initial-boundary-value problems,”* Journal of Computational Physics*, Vol. 268, pp. 17-38, 2014.