C. Schroeder, A. Stomakhin, R. Howes and J. Teran.
A Second Order Virtual Node Algorithm for
Navier-Stokes Flow Problems with Interfacial
Forces and Discontinuous Material Properties.
Journal of Computational Physics 2014.
We present a numerical method for the solution of the
Navier-Stokes equations in three dimensions that handles
interfacial discontinuities due to singular forces and
discontinuous fluid properties such as viscosity and
density. We show that this also allows for the enforcement
of normal stress and velocity boundary conditions on
irregular domains. The method provides treatment of fluid
inertia as well as a new discretization of jump and boundary
conditions that accurately resolves null modes in both two
and three dimensions. We discretize the equations using an
embedded approach on a uniform MAC grid to yield discretely
divergence-free velocities that are second order
accurate. We maintain our interface using the level set
method or, when more appropriate, the particle level set
method. We show how to implement Dirichlet (known velocity),
Neumann (known normal stress), and slip velocity boundary
conditions as special cases of our interface
representation. The method leads to a discrete, symmetric
KKT system for velocities, pressures, and Lagrange
multipliers. We also present a novel simplification to the
standard combination of the second order semi-Lagrangian and
BDF schemes for discretizing the inertial terms. Numerical
results indicate second order spatial accuracy for the
velocities (L-inf and L2) and first order for the pressure (in
L-inf, second order in L2). Our temporal discretization is
also second order accurate.