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Volume 30, Issue 1
A High-Order Discontinuous Galerkin Solver for Helically Symmetric Flows

Dominik Dierkes, Florian Kummer & Dominik Plümacher

Commun. Comput. Phys., 30 (2021), pp. 288-320.

Published online: 2021-04

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  • Abstract

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.

  • AMS Subject Headings

76T99, 35J75

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COPYRIGHT: © Global Science Press

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@Article{CiCP-30-288, author = {Dierkes , DominikKummer , 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 = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0204}, url = {http://global-sci.org/intro/article_detail/cicp/18882.html} }
TY - JOUR T1 - A High-Order Discontinuous Galerkin Solver for Helically Symmetric Flows AU - Dierkes , Dominik AU - Kummer , Florian AU - Plümacher , Dominik JO - Communications in Computational Physics VL - 1 SP - 288 EP - 320 PY - 2021 DA - 2021/04 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2019-0204 UR - https://global-sci.org/intro/article_detail/cicp/18882.html KW - Discontinuous Galerkin (DG), helical flows. AB -

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.

Dominik Dierkes, Florian Kummer & Dominik Plümacher. (2021). A High-Order Discontinuous Galerkin Solver for Helically Symmetric Flows. Communications in Computational Physics. 30 (1). 288-320. doi:10.4208/cicp.OA-2019-0204
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