A finite-difference scheme for three-dimensional incompressible flows in spherical coordinates

In this study we have developed a flexible and efficient numerical scheme for the simulation of three-dimensional incompressible flows in spherical coordinates. The main idea, inspired by a similar strategy as (Verzicco, R., Orlandi, P., 1996, A Finite-Difference Scheme for Three-Dimensional Incompressible Flows in Cylindrical Coordinates) for cylindrical coordinates, consists of a change of variables combined with a discretization on a staggered mesh and the special treatment of few discrete terms that remove the singularities of the Navier-Stokes equations at the sphere centre and along the polar axis. This new method alleviates also the time step restrictions introduced by the discretization around the polar axis while the sphere centre still yields strong limitations, although only in very unfavourable flow configurations. The scheme is second-order accurate in space and is verified and validated by computing numerical examples that are compared with similar results produced by other codes or available from the literature. The method can cope with flows evolving in the whole sphere, in a spherical shell and in a sector without any change and, thanks to the flexibility of finite-differences, it can employ generic mesh stretching (in two of the three directions) and complex boundary conditions.

[1]  Geert Brethouwer,et al.  A numerical investigation on the effect of the inflow conditions on the self-similar region of a round jet , 1998 .

[2]  Paolo Orlandi,et al.  A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage , 1994, Journal of Fluid Mechanics.

[3]  A. Babakov,et al.  Mathematical Simulation of Vortex Structures in Rapidly Rotating Astrophysical Objects , 2018, Computational Mathematics and Mathematical Physics.

[4]  F. Busse.  Patterns of convection in spherical shells , 1975, Journal of Fluid Mechanics.

[5]  M. J. P. Cullen,et al.  Current Progress and Prospects in Numerical Techniques for Weather Prediction Models , 1983 .

[6]  Transient natural convection heat transfer between concentric spheres , 1993 .

[7]  A. D. Young,et al.  An Introduction to Fluid Mechanics , 1968 .

[8]  Johannes Wicht.  Inner-core conductivity in numerical dynamo simulations , 2002 .

[9]  D. Kwak,et al.  A fractional step solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems , 1991 .

[10]  Franco Auteri,et al.  Navier-Stokes spectral solver in a sphere , 2009, J. Comput. Phys..

[11]  R. F. Warming,et al.  An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. [application to Eulerian gasdynamic equations , 1976 .

[12]  P. Gilman.  Linear Simulations of Boussinesq Convection in a Deep Rotating Spherical Shell , 1975 .

[13]  R. Verzicco,et al.  A Finite-Difference Scheme for Three-Dimensional Incompressible Flows in Cylindrical Coordinates , 1996 .

[14]  T. Colonius,et al.  On a transitional and turbulent natural convection in spherical shells , 2013 .

[15]  A. Tilgner,et al.  Spectral methods for the simulation of incompressible flows in spherical shells , 1999 .

[16]  Paolo Orlandi,et al.  Vortex rings impinging on walls: axisymmetric and three-dimensional simulations , 1993, Journal of Fluid Mechanics.

[17]  Bengt Fornberg,et al.  A Pseudospectral Approach for Polar and Spherical Geometries , 1995, SIAM J. Sci. Comput..

[18]  P. Gilman.  Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell-i , 1977 .

[19]  Keisuke Araki,et al.  Flow structure of thermal convection in a rotating spherical shell , 1996 .

[20]  P. Swarztrauber.  A direct Method for the Discrete Solution of Separable Elliptic Equations , 1974 .

[21]  Gareth P. Williams,et al.  Dynamics of a Convectively Unstable Atmosphere.: Jupiter? , 1973 .

[22]  Johannes Wicht,et al.  Turbulent Rayleigh–Bénard convection in spherical shells , 2015, Journal of Fluid Mechanics.

[23]  Jerry Westerweel,et al.  Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment , 1994, Journal of Fluid Mechanics.

[24]  Todd D. Ringler,et al.  Climate modeling with spherical geodesic grids , 2002, Comput. Sci. Eng..

[25]  S. Kida,et al.  Helical Flow Structure in a Precessing Sphere , 2008 .

[26]  T. Colonius,et al.  Numerical Treatment of Polar Coordinate Singularities , 2000 .

[27]  Detlef Lohse,et al.  Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection , 2009, Journal of Fluid Mechanics.

[28]  Weiming Sha,et al.  An Accurate Second-Order Approximation Factorization Method for Time-Dependent Incompressible Navier-Stokes Equations in Spherical Polar Coordinates , 1998 .

[29]  Jonathan M. Aurnou,et al.  The magnetic structure of convection-driven numerical dynamos , 2008 .

[30]  U. Ziegler,et al.  A semi-discrete central scheme for magnetohydrodynamics on orthogonal-curvilinear grids , 2011, J. Comput. Phys..

[31]  K. Watanabe,et al.  Simulation study of a magnetohydrodynamic dynamo : convection in a rotating spherical shell , 1993 .

[32]  Gary A. Glatzmaier,et al.  Numerical Simulations of Stellar Convective Dynamos. I. The Model and Method , 1984 .

[33]  Christian Klingenberg,et al.  Numerical Methods for Astrophysics , 2017 .

[34]  Numerical Simulation of Buoyancy-Induced Turbulent Flow Between Two Concentric Isothermal Spheres , 2010 .

[35]  P. Tackley,et al.  Focussing of stress by continents in 3D spherical mantle convection with self‐consistent plate tectonics , 2011 .

[36]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[37]  M. J. M. Hill,et al.  On a Spherical Vortex , 1894 .

[38]  P. Gresho.  Incompressible Fluid Dynamics: Some Fundamental Formulation Issues , 1991 .

[39]  Roberto Verzicco,et al.  A pencil distributed finite difference code for strongly turbulent wall-bounded flows , 2015, 1501.01247.

[40]  Donald L. Turcotte,et al.  Mantle Convection in the Earth and Planets , 2001 .

[41]  J. Kleimann,et al.  The CRONOS Code for Astrophysical Magnetohydrodynamics , 2018, The Astrophysical Journal Supplement Series.

[42]  F. Jülicher,et al.  Role of hydrodynamic flows in chemically driven droplet division , 2018, New Journal of Physics.

[43]  Harald Kruggel-Emden,et al.  A novel method for the calculation of particle heat conduction and resolved 3D wall heat transfer for the CFD/DEM approach , 2018, Powder Technology.

[44]  A. W. Vreman.  A staggered overset grid method for resolved simulation of incompressible flow around moving spheres , 2017, J. Comput. Phys..

[45]  S. Kida,et al.  A turbulent ring and dynamo in a precessing sphere , 2011 .

[46]  John K. Eaton,et al.  Fully resolved simulations of particle-turbulence interaction , 2003, Journal of Fluid Mechanics.

[47]  R. Johnson,et al.  Steady Large-Scale Ocean Flows in Spherical Coordinates , 2018, Oceanography.

[48]  Parviz Moin,et al.  On the suitability of second-order accurate discretizations for turbulent flow simulations , 2016 .

[49]  John R. Rice,et al.  Tensor product analysis of partial difference equations , 1964 .

[50]  Paolo Orlandi,et al.  Numerical simulations of transitional axisymmetric coaxial jets , 1996 .

[51]  C. Chiu,et al.  Transient natural convection heat transfer between concentric and vertically eccentric spheres , 1996 .