PROTEUS: A coupled iterative force-correction immersed-boundary cascaded lattice Boltzmann solver for moving and deformable boundary applications

Abstract Many realistic fluid flow problems are characterised by high Reynolds numbers and complex moving or deformable geometries. In our previous study, we presented a novel coupling between an iterative force-correction immersed boundary and a multi-domain cascaded lattice Boltzmann method, Falagkaris et al., and investigated flows around rigid bodies at Reynolds numbers up to 1 0 5 . Here, we extend its application to flows around moving and deformable bodies with prescribed motions. Emphasis is given on the influence of the internal mass on the computation of the aerodynamic forces including deforming boundary applications where the rigid body approximation is no longer valid. Both the rigid body and the internal Lagrangian points approximations are examined. The resulting solver has been applied to viscous flows around an in-line oscillating cylinder, a pitching foil, a plunging SD7003 airfoil and a plunging and flapping NACA-0014 airfoil. Good agreement with experimental results and other numerical schemes has been obtained. It is shown that the internal Lagrangian points approximation accurately captures the internal mass effects in linear and angular motions, as well as in deforming motions, at Reynolds numbers up to 4 ⋅ 1 0 4 . In all cases, the aerodynamic loads are significantly affected by the internal fluid forces.

[1]  Pierre Sagaut,et al.  Analysis of the absorbing layers for the weakly-compressible lattice Boltzmann methods , 2013, J. Comput. Phys..

[2]  Bastien Chopard,et al.  Lattice Boltzmann method with regularized pre-collision distribution functions , 2006, Math. Comput. Simul..

[3]  Petros Koumoutsakos,et al.  An immersed boundary-lattice-Boltzmann method for the simulation of the flow past an impulsively started cylinder , 2008, J. Comput. Phys..

[4]  Y. Hassan,et al.  A comparative study of direct‐forcing immersed boundary‐lattice Boltzmann methods for stationary complex boundaries , 2011 .

[5]  D. Kandhai,et al.  A generic, mass conservative local grid refinement technique for lattice‐Boltzmann schemes , 2006 .

[6]  Haecheon Choi,et al.  Immersed boundary method for flow around an arbitrarily moving body , 2006, J. Comput. Phys..

[7]  B. Chopard,et al.  Theory and applications of an alternative lattice Boltzmann grid refinement algorithm. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  W. G. Price,et al.  Low‐Reynolds‐number flow around an oscillating circular cylinder using a cell viscousboundary element method , 2001 .

[9]  T. Inamuro,et al.  Effect of internal mass in the simulation of a moving body by the immersed boundary method , 2011 .

[10]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[11]  Yongguang Cheng,et al.  Introducing unsteady non‐uniform source terms into the lattice Boltzmann model , 2008 .

[12]  Yan Peng,et al.  A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows , 2006 .

[13]  Q. Zou,et al.  On pressure and velocity boundary conditions for the lattice Boltzmann BGK model , 1995, comp-gas/9611001.

[14]  M. Uhlmann.  An immersed boundary method with direct forcing for the simulation of particulate flows , 2005, 1809.08170.

[15]  R. Glowinski,et al.  A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow , 2001 .

[16]  Takaji Inamuro,et al.  Flow between parallel walls containing the lines of neutrally buoyant circular cylinders , 2000 .

[17]  Ismail H. Tuncer,et al.  Thrust Generation Caused by Flapping Airfoils in a Biplane Configuration , 2003 .

[18]  Michael S. Triantafyllou,et al.  Forces on oscillating foils for propulsion and maneuvering , 2003 .

[19]  Orestis Malaspinas,et al.  Advances in multi-domain lattice Boltzmann grid refinement , 2012, J. Comput. Phys..

[20]  O. Filippova,et al.  Grid Refinement for Lattice-BGK Models , 1998 .

[21]  Santosh Ansumali,et al.  Single relaxation time model for entropic lattice Boltzmann methods. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  D. Wolf-Gladrow.  Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction , 2000 .

[23]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[24]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[25]  Yuan-Cheng Fung,et al.  An introduction to the theory of aeroelasticity , 1955 .

[26]  J. Korvink,et al.  Cascaded digital lattice Boltzmann automata for high Reynolds number flow. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Rolf Radespiel,et al.  Numerical and Experimental Flow Analysis of Moving Airfoils with Laminar Separation Bubbles , 2006 .

[28]  Luoding Zhu,et al.  Accuracy improvement of the immersed boundary–lattice Boltzmann coupling scheme by iterative force correction , 2016 .

[29]  B. Shizgal,et al.  Generalized Lattice-Boltzmann Equations , 1994 .

[30]  Xiaolei Yang,et al.  A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations , 2009, J. Comput. Phys..

[31]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[32]  Pengzhi Lin,et al.  Calculation of hydrodynamic forces acting on a submerged moving object using immersed boundary method , 2009 .

[33]  Sanjoy Banerjee,et al.  Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[35]  I. Karlin,et al.  Stabilization of the lattice boltzmann method by the H theorem: A numerical test , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[36]  Ruifeng Yuan,et al.  An immersed-boundary method based on the gas kinetic BGK scheme for incompressible viscous flow , 2015, J. Comput. Phys..

[37]  Luoding Zhu,et al.  Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGK Model , 2014 .

[38]  A. Ladd,et al.  Lattice-Boltzmann Simulations of Particle-Fluid Suspensions , 2001 .

[39]  J. Edwards,et al.  Computation vs. Experiment for High-Frequency Low-Reynolds Number Airfoil Pitch and Plunge , 2008 .

[40]  Marshall C. Galbraith,et al.  Implicit Large Eddy Simulation of Low-Reynolds-Number Transitional Flow Past the SD7003 Airfoil , 2010 .

[41]  P. Asinari.  Generalized local equilibrium in the cascaded lattice Boltzmann method. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  B. Shi,et al.  Discrete lattice effects on the forcing term in the lattice Boltzmann method. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Jung Il Choi,et al.  An immersed boundary method for complex incompressible flows , 2007, J. Comput. Phys..

[44]  Yu-Wei Chang,et al.  Simulating flows with moving rigid boundary using immersed-boundary method , 2010 .

[45]  J. Wu,et al.  Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications , 2009 .

[46]  E. Balaras.  Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations , 2004 .

[47]  Marcelo Reggio,et al.  An implementation of the Spalart-Allmaras turbulence model in a multi-domain lattice Boltzmann method for solving turbulent airfoil flows , 2015, Comput. Math. Appl..

[48]  W. Shyy,et al.  Viscous flow computations with the method of lattice Boltzmann equation , 2003 .

[49]  R. Glowinski,et al.  A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows , 2000 .

[50]  Weixing Yuan,et al.  An Investigation of Low-Reynolds-number Flows past Airfoils , 2005 .

[51]  J. Korvink,et al.  A factorized central moment lattice Boltzmann method , 2009 .

[52]  Z. Feng,et al.  Robust treatment of no-slip boundary condition and velocity updating for the lattice-Boltzmann simulation of particulate flows , 2009 .

[53]  J.-M. Miao,et al.  Effect of flexure on aerodynamic propulsive efficiency of flapping flexible airfoil , 2006 .

[54]  Qisu Zou,et al.  N ov 1 99 6 On pressure and velocity flow boundary conditions and bounceback for the lattice Boltzmann BGK model , 2008 .

[55]  F. Durst,et al.  Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan–Carpenter numbers , 1998, Journal of Fluid Mechanics.

[56]  J. Aider,et al.  A model for the symmetry breaking of the reverse Bénard–von Kármán vortex street produced by a flapping foil , 2008, Journal of Fluid Mechanics.

[57]  Manfred Krafczyk,et al.  Second order interpolation of the flow field in the lattice Boltzmann method , 2009, Comput. Math. Appl..

[58]  Takaji Inamuro,et al.  Lattice Boltzmann methods for moving boundary flows , 2012 .

[59]  S. Schwarzer,et al.  Navier-Stokes simulation with constraint forces: finite-difference method for particle-laden flows and complex geometries. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[60]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[61]  Martin Geier,et al.  The cumulant lattice Boltzmann equation in three dimensions: Theory and validation , 2015, Comput. Math. Appl..

[62]  Miguel R. Visbal,et al.  High-Fidelity Simulation of Transitional Flows Past a Plunging Airfoil , 2009 .

[63]  Martin Geier,et al.  Multi-thread implementations of the lattice Boltzmann method on non-uniform grids for CPUs and GPUs , 2011, Comput. Math. Appl..

[64]  J. R. Morison,et al.  The Force Exerted by Surface Waves on Piles , 1950 .

[65]  Yoshio Sone,et al.  Kinetic Theory and Fluid Dynamics , 2002 .

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

[67]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[68]  Jos Derksen,et al.  Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity , 2002 .

[69]  Ignazio Maria Viola,et al.  PROTEUS: A coupled iterative force-correction immersed-boundary multi-domain cascaded lattice Boltzmann solver , 2017, Comput. Math. Appl..

[70]  M. Junk,et al.  Asymptotic analysis of the lattice Boltzmann equation , 2005 .

[71]  Z. Feng,et al.  The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems , 2004 .

[72]  Laurens E. Howle,et al.  Spring stiffness influence on an oscillating propulsor , 2003 .

[73]  C. Eloy,et al.  Transitions in the wake of a flapping foil. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.