@Article{CiCP-30-1,
author = {Dominik, Dierkes and Kummer, Florian and Plümacher, Dominik},
title = {A High-Order Discontinuous Galerkin Solver for Helically Symmetric Flows},
journal = {Communications in Computational Physics},
year = {2021},
volume = {30},
number = {1},
pages = {288--320},
abstract = {<p style="text-align: justify;">We present a high-order discontinuous Galerkin (DG) scheme to solve the
system of helically symmetric Navier-Stokes equations which are discussed in [28].
In particular, we discretize the helically reduced Navier-Stokes equations emerging
from a reduction of the independent variables such that the remaining variables are: $t$, $r$, $ξ$ with $ξ=az+bϕ$, where $r$, $ϕ$, $z$ are common cylindrical coordinates and $t$ the
time. Beside this, all three velocity components are kept non-zero. A new non-singular
coordinate $η$ is introduced which ensures that a mapping of helical solutions into the
three-dimensional space is well defined. Using that, periodicity conditions for the
helical frame as well as uniqueness conditions at the centerline axis $r=0$ are derived. In
the sector near the axis of the computational domain a change of the polynomial basis
is implemented such that all physical quantities are uniquely defined at the centerline.<br/>For the temporal integration, we present a semi-explicit scheme of third order
where the full spatial operator is split into a Stokes operator which is discretized
implicitly and an operator for the nonlinear terms which is treated explicitly. Computations are conducted for a cylindrical shell, excluding the centerline axis, and for
the full cylindrical domain, where the centerline is included. In all cases we obtain the
convergence rates of order $\mathcal{O}(h^{k+1})$ that are expected from DG theory.<br/>In addition to the first DG discretization of the system of helically invariant Navier-Stokes equations, the treatment of the central axis, the resulting reduction of the DG
space, and the simultaneous use of a semi-explicit time stepper are of particular novelty.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2019-0204},
url = {https://global-sci.com/article/79569/a-high-order-discontinuous-galerkin-solver-for-helically-symmetric-flows}
}