Assessment of a high-order discontinuous Galerkin method for incompressible three-dimensional Navier–Stokes equations: Benchmark results for the flow past a sphere up to Re = 500

Abstract This paper deals with the implementation of the high-order Discontinuous Galerkin (DG) artificial compressibility flux method [1] into a three-dimensional incompressible Navier–Stokes (INS) solver. The method is fully implicit in time and its distinguishing feature is the formulation of the inviscid interface flux, which is based on the solution of the Riemann problem associated with a local artificial compressibility-like perturbation of the equations. The code has been tested on a wide range of flow regimes considering the flow past a sphere at moderate Reynolds numbers. In order to asses the code reliability and its accuracy in space, up to the sixth order polynomial approximation, and in time, up to the fourth order, both steady (Re = 20, 200, 250) and unsteady (Re = 300, 500) problems have been approached. With the largest Reynolds number here considered the flow exhibits a complex behavior, even if it still laminar, since undergoes the transition from a regular to an almost chaotic system. For this problem, for which the flow field characteristics are not completely well established and no many data are available in the present literature, detailed results are reported in this paper.

[1]  K. S. Yeo,et al.  A SVD-GFD scheme for computing 3D incompressible viscous fluid flows , 2008 .

[2]  Chi-Wang Shu,et al.  A High-Order Discontinuous Galerkin Method for 2D Incompressible Flows , 2000 .

[3]  E. Achenbach,et al.  Vortex shedding from spheres , 1974, Journal of Fluid Mechanics.

[4]  Sungsu Lee,et al.  A numerical study of the unsteady wake behind a sphere in a uniform flow at moderate Reynolds numbers , 2000 .

[5]  C. Brücker.  SPATIO-TEMPORAL RECONSTRUCTION OF VORTEX DYNAMICS IN AXISYMMETRIC WAKES , 2001 .

[6]  Hiroshi Sakamoto,et al.  The formation mechanism and shedding frequency of vortices from a sphere in uniform shear flow , 1995, Journal of Fluid Mechanics.

[7]  R. Mittal.  A Fourier–Chebyshev spectral collocation method for simulating flow past spheres and spheroids , 1999 .

[8]  V. C. Patel,et al.  Flow past a sphere up to a Reynolds number of 300 , 1999, Journal of Fluid Mechanics.

[9]  William W. Willmarth,et al.  Some experimental results on sphere and disk drag , 1971 .

[10]  A. J. Baker,et al.  A stiffly-stable implicit Runge-Kutta algorithm for CFD applications , 1988 .

[11]  Lawrence F. Shampine,et al.  Implementation of Rosenbrock Methods , 1982, TOMS.

[12]  E. Achenbach,et al.  Experiments on the flow past spheres at very high Reynolds numbers , 1972, Journal of Fluid Mechanics.

[13]  Roy L. Bishop,et al.  Wakes in Liquid‐Liquid Systems , 1961 .

[14]  F. Bassi,et al.  High-order discontinuous Galerkin solutions of three-dimensional incompressible RANS equations , 2013 .

[15]  K. Squires,et al.  LES and DES investigations of turbulent flow over a sphere , 2000 .

[16]  Daniele A. Di Pietro,et al.  A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure , 2011, J. Comput. Phys..

[17]  Ronald Cools,et al.  An encyclopaedia of cubature formulas , 2003, J. Complex..

[18]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

[19]  Guido Kanschat,et al.  A locally conservative LDG method for the incompressible Navier-Stokes equations , 2004, Math. Comput..

[20]  Jinhee Jeong,et al.  On the identification of a vortex , 1995, Journal of Fluid Mechanics.

[21]  Jens Lang,et al.  ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems , 2000 .

[22]  H. Schlichting.  Boundary Layer Theory , 1955 .

[23]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations , 2011, J. Comput. Phys..

[24]  Rajat Mittal,et al.  Vortex dynamics in the sphere wake , 1999 .

[25]  H. Sakamoto,et al.  A STUDY ON VORTEX SHEDDING FROM SPHERES IN A UNIFORM FLOW , 1990 .

[26]  D. Hartmann,et al.  A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids , 2011 .

[27]  C. Ross Ethier,et al.  A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations , 2007, J. Comput. Phys..

[28]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[29]  Guido Kanschat,et al.  The local discontinuous Galerkin method for linearized incompressible fluid flow: a review , 2005 .

[30]  L. Schouveiler,et al.  Self-sustained oscillations in the wake of a sphere , 2002 .

[31]  Jungwoo Kim,et al.  An immersed-boundary finite-volume method for simulations of flow in complex geometries , 2001 .

[32]  Michael S. Warren,et al.  Vortex Methods for Direct Numerical Simulation of Three-Dimensional Bluff Body Flows , 2002 .

[33]  S. Taneda.  Experimental Investigation of the Wake behind a Sphere at Low Reynolds Numbers , 1956 .

[34]  Andrea Crivellini,et al.  An implicit matrix-free Discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations , 2011 .

[35]  M. Tabata,et al.  A precise computation of drag coefficients of a sphere , 1998 .

[36]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[37]  Rajat Mittal,et al.  Symmetry Properties of the Transitional Sphere Wake , 2002 .

[38]  Valerio D'Alessandro,et al.  A Spalart-Allmaras turbulence model implementation in a discontinuous Galerkin solver for incompressible flows , 2013, J. Comput. Phys..

[39]  Andrea Crivellini,et al.  An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations , 2006, J. Comput. Phys..

[40]  Guido Kanschat,et al.  Local Discontinuous Galerkin Methods for the Stokes System , 2002, SIAM J. Numer. Anal..

[41]  S. Rebay,et al.  An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows , 2007 .

[42]  Gerd Steinebach,et al.  Order-reduction of ROW-methods for DAEs and method of lines applications , 1995 .

[43]  Wolfgang Schröder,et al.  A lattice-Boltzmann method with hierarchically refined meshes , 2013 .

[44]  Esteban Ferrer,et al.  A high order Discontinuous Galerkin - Fourier incompressible 3D Navier-Stokes solver with rotating sliding meshes , 2012, J. Comput. Phys..

[45]  S. Orszag,et al.  Numerical investigation of transitional and weak turbulent flow past a sphere , 2000, Journal of Fluid Mechanics.