A dimensionally split Cartesian cut cell method for the compressible Navier-Stokes equations

Abstract We present a dimensionally split method for computing solutions to the compressible Navier–Stokes equations on Cartesian cut cell meshes. The method is globally second order accurate in the L 1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a number of test problems ranging from the nearly incompressible to the highly compressible flow regimes. All the computed results show good agreement with reference results from theory, experiment and previous numerical studies. To the best of our knowledge, this is the first presentation of a dimensionally split cut cell method for the compressible Navier–Stokes equations in the literature.

[1]  Nikolaus A. Adams,et al.  A conservative immersed interface method for Large-Eddy Simulation of incompressible flows , 2010, J. Comput. Phys..

[2]  Nikolaus A. Adams,et al.  A conservative interface method for compressible flows , 2006, J. Comput. Phys..

[3]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[4]  P. Colella,et al.  A cartesian grid embedded boundary method for the compressible Navier–Stokes equations , 2013 .

[5]  Kazuhiro Nakahashi,et al.  Improved Formulation for Geometric Properties of Arbitrary Polyhedra , 2003 .

[6]  C. Farhat,et al.  An Embedded Boundary Method for Viscous Fluid/Structure Interaction Problems and Application to Flexible Flapping Wings , 2012 .

[7]  M. Berger,et al.  Progress Towards a Cartesian Cut-Cell Method for Viscous Compressible Flow , 2012 .

[8]  C. Wieselsberger.  New Data on the Laws of Fluid Resistance , 1922 .

[9]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[10]  A. B. Bailey,et al.  Sphere Drag Coefficients for a Broad Range of Mach and Reynolds Numbers , 1972 .

[11]  Matthias Meinke,et al.  A Cartesian cut-cell method for sharp moving boundaries , 2011 .

[12]  N. Nikiforakis,et al.  Well-balanced compressible cut-cell simulation of atmospheric flow , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[13]  F. Capizzano.  Turbulent Wall Model for Immersed Boundary Methods , 2011 .

[14]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[15]  A. S. Grove,et al.  An experimental investigation of the steady separated flow past a circular cylinder , 1964, Journal of Fluid Mechanics.

[16]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[17]  Farrokh Najmabadi,et al.  An embedded boundary method for viscous, conducting compressible flow , 2004, J. Comput. Phys..

[18]  J. Ferziger,et al.  A ghost-cell immersed boundary method for flow in complex geometry , 2002 .

[19]  Wolfgang Schröder,et al.  An efficient conservative cut-cell method for rigid bodies interacting with viscous compressible flows , 2016, J. Comput. Phys..

[20]  I. I. Glass,et al.  A numerical study of oblique shock-wave reflections with experimental comparisons , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  Randall J. LeVeque,et al.  A High-Resolution Rotated Grid Method for Conservation Laws with Embedded Geometries , 2005, SIAM J. Sci. Comput..

[22]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[23]  D. Tritton.  Experiments on the flow past a circular cylinder at low Reynolds numbers , 1959, Journal of Fluid Mechanics.

[24]  Wolfgang Schröder,et al.  An accurate moving boundary formulation in cut-cell methods , 2013, J. Comput. Phys..

[25]  S. Armfield,et al.  A representation of curved boundaries for the solution of the Navier-Stokes equations on a staggered three-dimensional Cartesian grid , 2003 .

[26]  Marsha Berger,et al.  Cut Cells: Meshes and Solvers , 2017 .

[27]  N. Adams,et al.  Wall modeling for implicit large-eddy simulation and immersed-interface methods , 2014 .

[28]  Suresh Menon,et al.  A high-order adaptive Cartesian cut-cell method for simulation of compressible viscous flow over immersed bodies , 2016, J. Comput. Phys..

[29]  M. Lai,et al.  An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity , 2000 .

[30]  Phillip Colella,et al.  A Cartesian grid embedded boundary method for hyperbolic conservation laws , 2006 .

[31]  M. D. Salas,et al.  Euler calculations for multielement airfoils using Cartesian grids , 1986 .

[32]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[33]  M. Al-Marouf,et al.  A versatile embedded boundary adaptive mesh method for compressible flow in complex geometry , 2017, J. Comput. Phys..

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

[35]  M. Berger,et al.  An ODE-based wall model for turbulent flow simulations , 2017 .

[36]  Carlos Pantano,et al.  A Cartesian-based embedded geometry technique with adaptive high-order finite differences for compressible flow around complex geometries , 2014, J. Comput. Phys..

[37]  Marsha J. Berger,et al.  A Simplified h-box Method for Embedded Boundary Grids , 2012, SIAM J. Sci. Comput..

[38]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[39]  Nikos Nikiforakis,et al.  A dimensionally split Cartesian cut cell method for hyperbolic conservation laws , 2017, J. Comput. Phys..

[40]  S. Mauch.  A Fast Algorithm for Computing the Closest Point and Distance Transform , 2000 .

[41]  A. P. Krasil'shchikov,et al.  Experimental study of sphere aerodynamic characteristics in free flight up to M ~ 15 , 1968 .

[42]  J. Quirk.  An alternative to unstructured grids for computing gas dynamic flows around arbitrarily complex two-dimensional bodies , 1994 .