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 eq...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,ξwithξ=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 coordinateηis 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 aswell 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 splitted 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 O(hk+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.展开更多
文摘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,ξwithξ=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 coordinateηis 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 aswell 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 splitted 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 O(hk+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.
