## Communications in Applied Mathematics and Computational Science

### An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries

#### Abstract

We present an adaptive, finite volume algorithm to solve the incompressible Navier–Stokes equations in complex geometries. The algorithm is based on the embedded boundary method, in which finite volume approximations are used to discretize the solution in cut cells that result from intersecting the irregular boundary with a structured Cartesian grid. This approach is conservative and reduces to a standard finite difference method in grid cells away from the boundary. We solve the incompressible flow equations using a predictor-corrector formulation. Hyperbolic advection terms are obtained by higher-order upwinding without the use of extrapolated data in covered cells. The small-cell stability problem associated with explicit embedded boundary methods for hyperbolic systems is avoided by the use of a volume-weighted scheme in the advection step and is consistent with construction of the right-hand side of the elliptic solvers. The Helmholtz equations resulting from viscous source terms are advanced in time by the Crank–Nicolson method, which reduces solver runtime compared to other second-order time integrators by a half. Incompressibility is enforced by a second-order approximate projection method that makes use of a new conservative cell-centered gradient in cut cells that is consistent with the volume-weighted scheme. The algorithm is also capable of block structured adaptive mesh refinement to increase spatial resolution dynamically in regions of interest. The resulting overall method is second-order accurate for sufficiently smooth problems. In addition, the algorithm is implemented in a high-performance computing framework and can perform structured-grid fluid dynamics calculations at unprecedented scale and resolution, up to 262,144 processor cores. We demonstrate robustness and performance of the algorithm by simulating incompressible flow for a wide range of Reynolds numbers in two and three dimensions: Stokes and low Reynolds number flows in both constructed and image data geometries ($Re ≪ 1$ to $Re = 1$), flow past a cylinder ($Re = 300$), flow past a sphere ($Re = 600$) and turbulent flow in a contraction ($Re = 6300$).

#### Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 10, Number 1 (2015), 43-82.

Dates
Revised: 29 October 2014
Accepted: 7 December 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.camcos/1510858401

Digital Object Identifier
doi:10.2140/camcos.2015.10.43

Mathematical Reviews number (MathSciNet)
MR3327727

Zentralblatt MATH identifier
06425515

#### Citation

Trebotich, David; Graves, Daniel. An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries. Commun. Appl. Math. Comput. Sci. 10 (2015), no. 1, 43--82. doi:10.2140/camcos.2015.10.43. https://projecteuclid.org/euclid.camcos/1510858401

#### References

• M. J. Aftosmis, M. J. Berger, and J. E. Melton, Robust and efficient Cartesian mesh generation for component-base geometry, AIAA J. 36 (1998), no. 6, 952–960.
• A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, and M. L. Welcome, A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations, J. Comput. Phys. 142 (1998), no. 1, 1–46. \xxMR99k:76096 \xxZBL0933.76055
• A. S. Almgren, J. B. Bell, P. Colella, and T. Marthaler, A Cartesian grid projection method for the incompressible Euler equations in complex geometries, SIAM J. Sci. Comput. 18 (1997), no. 5, 1289–1309. \xxMR98d:76112 \xxZBL0910.76040
• J. B. Bell, P. Colella, and H. M. Glaz, A second-order projection method for the incompressible Navier–Stokes equations, J. Comput. Phys. 85 (1989), no. 2, 257–283. \xxMR90i:76002 \xxZBL0681.76030
• J. B. Bell, P. Colella, and L. H. Howell, An efficient second-order projection method for viscous incompressible flow, 10th Computational Fluid Dynamics Conference (Honolulu, 1991), AIAA, 1991, pp. 360–367.
• J. B. Bell, P. Colella, and M. L. Welcome, Conservative front-tracking for inviscid compressible flow, 10th Computational Fluid Dynamics Conference (Honolulu, 1991), AIAA, 1991, pp. 814–822.
• M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys. 82 (1989), no. 1, 64–84. \xxZBL0665.76070
• M. J. Berger, C. Helzel, and R. J. LeVeque, $h$-box methods for the approximation of hyperbolic conservation laws on irregular grids, SIAM J. Numer. Anal. 41 (2003), no. 3, 893–918. \xxMR2004g:65103 \xxZBL1066.65082
• M. J. Berger and R. J. LeVeque, Stable boundary conditions for Cartesian grid calculations, Comput. Syst. Eng. 1 (1990), no. 2–4, 305–311.
• M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys. 53 (1984), no. 3, 484–512. \xxMR85h:65211 \xxZBL0536.65071
• I.-L. Chern and P. Colella, A conservative front-tracking method for hyperbolic conservation laws, Technical Report UCRL-97200, Lawrence Livermore National Laboratory, 1987.
• A. J. Chorin, Numerical solution of the Navier–Stokes equations, Math. Comp. 22 (1968), 745–762. \xxMR39 #3723 \xxZBL0198.50103
• M.-H. Chung, Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape, Computers and Fluids 35 (2006), no. 6, 607–623. \xxZBL1160.76369
• W. J. Coirier and K. G. Powell, An accuracy assessment of Cartesian-mesh approaches for the Euler equations, J. Comput. Phys. 117 (1995), no. 1, 121–131. \xxMR95k:76088 \xxZBL0817.76055
• P. Colella, D. T. Graves, B. J. Keen, and D. Modiano, A Cartesian grid embedded boundary method for hyperbolic conservation laws, J. Comput. Phys. 211 (2006), no. 1, 347–366. \xxMR2006i:65142 \xxZBL1120.65324
• P. Colella and D. P. Trebotich, Numerical simulation of incompressible viscous flow in deforming domains, Proc. Natl. Acad. Sci. USA 96 (1999), no. 10, 5378–5381. \xxMR2000a:76116 \xxZBL0938.76063
• R. P. Fedkiw, T. Aslam, B. Merriman, and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows $($the ghost fluid method$)$, J. Comput. Phys. 152 (1999), no. 2, 457–492. \xxMR2000c:76061 \xxZBL0957.76052
• R. P. Fedkiw and X.-D. Liu, The ghost fluid method for viscous flows, Innovative methods for numerical solutions of partial differential equations (Arcachon, France, 1998) (M. M. Hafez and J.-J. Chattot, eds.), World Sci. Publ., 2002, pp. 111–143. \xxMR1928585 \xxZBL1078.76586
• D. T. Graves, P. Colella, D. Modiano, J. Johnson, B. Sjogreen, and X. Gao, A Cartesian grid embedded boundary method for the compressible Navier–Stokes equations, Commun. Appl. Math. Comput. Sci. 8 (2013), no. 1, 99–122. \xxMR3143820 \xxZBL1282.76006
• C. Helzel, M. J. Berger, and R. J. LeVeque, A high-resolution rotated grid method for conservation laws with embedded geometries, SIAM J. Sci. Comput. 26 (2005), no. 3, 785–809. \xxMR2005i:35180 \xxZBL1074.35071
• H. Johansen and P. Colella, A Cartesian grid embedded boundary method for Poisson's equation on irregular domains, J. Comput. Phys. 147 (1998), no. 1, 60–85. \xxMR99m:65231 \xxZBL0923.65079
• G. E. Karniadakis and G. S. Triantafyllou, Three-dimensional dynamics and transition to turbulence in the wake of bluff objects, J. Fluid Mech. 238 (1992), 1–30. \xxZBL0754.76043
• B. Keen and S. Karni, A second order kinetic scheme for gas dynamics on arbitrary grids, J. Comput. Phys. 205 (2005), no. 1, 108–130. \xxMR2005k:76105 \xxZBL1087.76088
• J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier–Stokes equations, J. Comput. Phys. 59 (1985), no. 2, 308–323. \xxMR87a:76046 \xxZBL0582.76038
• M. P. Kirkpatrick, S. W. Armfield, and J. H. Kent, A representation of curved boundaries for the solution of the Navier–Stokes equations on a staggered three-dimensional Cartesian grid, J. Comput. Phys. 184 (2003), no. 1, 1–36. \xxZBL1118.76350
• M. F. Lai, A projection method for reacting flow in the zero Mach number limit, Ph.D. thesis, University of California, Berkeley, 1993. \xxMR2691063
• M.-C. Lai and C. S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. minus1pt Comput. minus1pt Phys. 160 (2000), no. minus1pt 2, 705–719. \xxMR2000m:76085 \xxZBL0954.76066
• R. J. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31 (1994), no. 4, 1019–1044. \xxMR95g:65139 \xxZBL0811.65083
• T. J. Ligocki, P. O. Schwartz, J. Percelay, and P. Colella, Embedded boundary grid generation using the divergence theorem, implicit functions, and constructive solid geometry, SciDAC 2008 (Seattle), J. Phys. Conf. Ser., no. 125, 2008.
• D. F. Martin and P. Colella, A cell-centered adaptive projection method for the incompressible Euler equations, J. Comput. Phys. 163 (2000), no. 2, 271–312. \xxMR2001g:76040 \xxZBL0991.76052
• D. F. Martin, P. Colella, and D. Graves, A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions, J. Comput. Phys. 227 (2008), no. 3, 1863–1886. \xxMR2009g:76085 \xxZBL1137.76040
• P. McCorquodale, P. Colella, and H. Johansen, A Cartesian grid embedded boundary method for the heat equation on irregular domains, J. Comput. Phys. 173 (2001), no. 2, 620–635. \xxMR2002h:80009 \xxZBL0991.65099
• M. K. McNutt, S. Chu, J. Lubchenco, T. Hunter, G. Dreyfus, S. A. Murawski, and D. M. Kennedy, Applications of science and engineering to quantify and control the Deepwater Horizon oil spill, Proc. Natl. Acad. Sci. USA 109 (2012), no. 50, 20222–20228.
• G. H. Miller and D. Trebotich, An embedded boundary method for the Navier–Stokes equations on a time-dependent domain, Commun. Appl. Math. Comput. Sci. 7 (2012), no. 1, 1–31. \xxMR2893419 \xxZBL1273.35215
• M. L. Minion, On the stability of Godunov-projection methods for incompressible flow, J. Comput. Phys. 123 (1996), no. 2, 435–449. \xxMR96j:76111 \xxZBL0848.76050
• H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18 (1964), no. 1, 1–18. \xxZBL0118.20501
• S. Molins, D. Trebotich, C. I. Steefel, and C. Shen, An investigation of the effect of pore scale flow on average geochemical reaction rates using direct numerical simulation, Water Resour. Res. 48 (2012), no. 3.
• S. Molins, D. Trebotich, L. Yang, J. B. Ajo-Franklin, T. J. Ligocki, C. Shen, and C. I. Steefel, Pore-scale controls on calcite dissolution rates from flow-through laboratory and numerical experiments, Environ. Sci. Technol. 48 (2014), no. 13, 7453–7460.
• C. M. Oldenburg, B. M. Freifeld, K. Pruess, L. Pan, S. Finsterle, and G. J. Moridis, Numerical simulations of the Macondo well blowout reveal strong control of oil flow by reservoir permeability and exsolution of gas, Proc. Natl. Acad. Sci. USA 109 (2012), no. 50, 20254–20259.
• R. B. Pember, J. B. Bell, P. Colella, W. Y. Crutchfield, and M. L. Welcome, An adaptive Cartesian grid method for unsteady compressible flow in irregular regions, J. Comput. Phys. 120 (1995), no. 2, 278–304. \xxMR96d:76081 \xxZBL0842.76056
• ––––, An adaptive Cartesian grid method for unsteady compressible flow in irregular regions, J. Comput. Phys. 120 (1995), no. 2, 278–304. \xxMR96d:76081 \xxZBL0842.76056
• C. S. Peskin, Flow patterns around heart valves: a numerical method, J. Comput. Phys. 10 (1972), no. 2, 252–271. \xxZBL0244.92002
• S. Popinet, Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. Comput. Phys. 190 (2003), no. 2, 572–600. \xxMR2013029 \xxZBL1076.76002
• J. J. Quirk, An alternative to unstructured grids for computing gas dynamics flows around arbitrarily complex two-dimensional bodies, Computers and Fluids 23 (1994), no. 1, 125–142. \xxZBL0788.76067
• P. O. Schwartz, J. Percelay, T. Ligocki, H. Johansen, D. T. Graves, P. Devendran, P. Colella, and E. Ateljevich, High-accuracy embedded boundary grid generation using the divergence theorem, preprint, 2015, To appear in Commun. Appl. Math. Comput. Sci.
• K. Shahbazi, P. F. Fischer, and C. R. Ethier, A high-order discontinuous Galerkin method for the unsteady incompressible Navier–Stokes equations, J. Comput. Phys. 222 (2007), no. 1, 391–407. \xxMR2007k:76097 \xxZBL1216.76034
• D. Trebotich, Simulation of biological flow and transport in complex geometries using embedded boundary/volume-of-fluid methods, SciDAC 2007 (Boston), J. Phys. Conf. Ser., no. 78, 2007.
• D. Trebotich, M. F. Adams, S. Molins, C. I. Steefel, and C. Shen, High-resolution simulation of pore-scale reactive transport processes associated with carbon sequestration, Comput. Sci. Eng. 16 (2014), no. 6, 22–31.
• D. Trebotich and P. Colella, A projection method for incompressible viscous flow on moving quadrilateral grids, J. Comput. Phys. 166 (2001), no. 2, 191–217. \xxMR2001m:76076 \xxZBL1030.76044
• D. Trebotich, P. Colella, G. Miller, A. Nonaka, T. Marshall, S. Gulati, and D. Liepmann, A numerical algorithm for complex biological flow in irregular microdevice geometries, Technical proceedings of the 2004 Nanotechnology Conference and Trade Show (Boston), vol. 2, Nano Science and Technology Institute, 2004, pp. 470–473.
• D. Trebotich, G. H. Miller, and M. D. Bybee, A penalty method to model particle interactions in DNA-laden flows, J. Nanosci. Nanotechnol. 8 (2008), no. 7, 3749–3756.
• D. Trebotich, G. H. Miller, P. Colella, D. T. Graves, D. F. Martin, and P. O. Schwartz, A tightly coupled particle-fluid model for DNA-laden flows in complex microscale geometries, Computational Fluid and Solid Mechanics 2005 (Cambridge, MA) (K. J. Bathe, ed.), Elsevier, 2005, pp. 1018–1022.
• D. Trebotich, B. Van Straalen, D. T. Graves, and P. Colella, Performance of embedded boundary methods for CFD with complex geometry, SciDAC 2008 (Seattle), J. Phys. Conf. Ser., no. 125, 2008.
• E. H. Twizell, A. B. Gumel, and M. A. Arigu, Second-order, $L\sb 0$-stable methods for the heat equation with time-dependent boundary conditions, Adv. Comput. Math. 6 (1996), no. 1, 333–352. \xxMR97m:65164 \xxZBL0872.65084
• J. van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 870–891. \xxMR87h:76008 \xxZBL0594.76023
• B. van Leer, Towards the ultimate conservative difference scheme, V: A second-order sequel to Godunov's method, J. Comput. Phys. 32 (1979), no. 1, 101–136.
• C. H. K. Williamson, Defining a universal and continuous Strouhal–Reynolds number relationship for the laminar vortex shedding of a circular cylinder, Phys. Fluids 31 (1988), no. 10, 2742–2744.
• ––––, The existence of two stages in the transition to three-dimensionality of a cylinder wake, Phys. Fluids 31 (1988), no. 11, 3165–3168.
• ––––, Vortex dynamics in the cylinder wake, Annual review of fluid mechanics (Palo Alto, CA) (J. L. Lumley, M. Van Dyke, and H. L. Reed, eds.), vol. 28, Annual Reviews, 1996, pp. 477–539. \xxMR96j:76051
• T. Ye, R. Mittal, H. S. Udaykumar, and W. Shyy, An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. Comput. Phys. 156 (1999), no. 2, 209–240. \xxZBL0957.76043
• H.-Q. Zhang, U. Fey, B. R. Noack, M. König, and H. Eckelmann, On the transition of the cylinder wake, Phys. Fluids 7 (1995), no. 4, 779–794.