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In this thesis helical flows are investigated, in which the fluid particles simultaneously perform a rotational as well as a translational motion and, thus, move along a helix. The special feature of such flows is that they are based on a dimensional reduction, i.e. the number of coordinates, used to describe the flow is reduced. Such a reduction is referred to as dimensional reduction, since each coordinate represents one spatial dimension. The present work is divided into an analytical and a numerical part. In the analytical part, a new time-dependent coordinate system is derived from the symmetries of the incompressible Navier-Stokes equations. New conservation laws for viscous and non-viscous helical flows could be found for this coordinate system, which are shown in this thesis and have been published in the article Dierkes and Oberlack (2017). Furthermore, we consider the classical, temporally constant helical coordinate system and derive two classes of new exact solutions of the helical symmetric, full time-dependent Navier-Stokes equations. The first class of solutions is based on the symmetries of the Navier-Stokes equations and hence are denoted as invariant solutions. The second class of solutions is based on a linearization of the Navier-Stokes equations using the so-called Beltrami condition, whereby the velocity and vorticity vectors are assumed to be parallel to each other. In the numerical part of the work, a solver for the simulation of helically symmetrical flows is developed using the discontinuous Galerkin (DG) method, in which the solution is approximated by high-order polynomials. Due to the fact that helical flows in most cases are periodically in the direction of the central axis of the helix, a periodicity condition for the helical coordinates is derived. A condition for the velocity and the pressure is formulated analogously to the procedure known from the literature for axisymmetric flows (cf. Khorrami et al., 1989). This ensures the uniqueness of these physical quantities at the central axis of the helix. In addition, we introduce a suitable function space and formulate the spatial and temporal discretization of the helically symmetric Navier-Stokes equations. For the temporal discretization, we use a third order semi-explicit method in which the spatial operator is split into an explicit and an implicit part. Using this, the computational effort for transient simulations has been reduced significantly. The correct implementation is verified by various test cases including the exact solutions which have been found before. It is further shown that the convergence rates that we expect from theory are achieved. Finally, the results of direct numerical simulations at high Reynolds numbers are performed which reveal the formation of vortices, Kelvin-Helmholtz instabilities and the temporal development of energy spectra for helically invariant flows.

A high-order discontinuous Galerkin solver and exact solutions for helically invariant flows