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.
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.
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.