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

Publication Date:
Research Org.:
Energy Frontier Research Centers (EFRC) (United States). Center for Nanoscale Control of Geologic CO2 (NCGC)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)
OSTI Identifier:
DOE Contract Number:
Resource Type:
Journal Article
Resource Relation:
Journal Name: Communications in Applied Mathematics and Computational Science; Journal Volume: 10; Journal Issue: 1; Related Information: NCGC partners with Lawrence Berkeley National Laboratory (lead); University of California, Davis; Lawrence Livermore National Laboratory; Massachusetts Institute of Technology; Ohio State University; Oak Ridge National Laboratory; Washington University, St. Louis
Country of Publication:
United States
bio-inspired, mechanical behavior, carbon sequestration

Citation Formats

Trebotich, David, and Graves, Daniel. An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries. United States: N. p., 2015. Web. doi:10.2140/camcos.2015.10.43.
Trebotich, David, & Graves, Daniel. An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries. United States. doi:10.2140/camcos.2015.10.43.
Trebotich, David, and Graves, Daniel. 2015. "An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries". United States. doi:10.2140/camcos.2015.10.43.
title = {An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries},
author = {Trebotich, David and Graves, Daniel},
abstractNote = {},
doi = {10.2140/camcos.2015.10.43},
journal = {Communications in Applied Mathematics and Computational Science},
number = 1,
volume = 10,
place = {United States},
year = 2015,
month = 1
  • A general multidomain decomposition is proposed for the numerical solution of the 2D incompressible stationary Navier-Stokes equations. The solution technique consists in a Chebyshev orthogonal collocation method preconditioned by a standard Galerkin finite element technique. The preconditioned system is then solved through a Richardson Procedure. The domain of interest is decomposed into quadrilaterals, curved when needed. A Gordon transfinite interpolation performs the curvilinear grid generation of the obtained simply-connected planar subdomains. The interface conditions, naturally incorporated into the finite element approach, relate neighbor subdomains through the normal jump of appropriate fluxes across internal boundaries, where an integral form of C[supmore » 1] continuity is consequently achieved at convergence of the iterative processes. The study of model Stokes problems demonstrates that the current method still behaves spectrally in distorted geometries. For curvilinear distortion, a loss of several orders of magnitude is observed in the solution accuracy even when the distortion is very limited. Finally, some results of flow Simulation in a constricted channel are proposed to illustrate the abilities of the present method to treat Navier-Stokes problems. 31 refs.« less
  • The discontinuous Petrov-Galerkin methodology with optimal test functions (DPG) of Demkowicz and Gopalakrishnan [18, 20] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. Whereas Bubnov-Galerkin methods use identical trial and test spaces, Petrov-Galerkin methods allow these function spaces to differ. In DPG, test functions are computed on the fly and are chosen to realize the supremum in the inf-sup condition; the method is equivalent to a minimum residual method. For well-posed problems with sufficiently regular solutions, DPG can be shown to converge at optimal rates—the inf-sup constants governing the convergence aremore » mesh-independent, and of the same order as those governing the continuous problem [48]. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. We employ DPG to solve the steady incompressible Navier-Stokes equations in two dimensions, building on previous work on the Stokes equations, and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh. We apply our approach to a manufactured solution due to Kovasznay as well as the lid-driven cavity flow, backward-facing step, and flow past a cylinder problems.« less
  • Cited by 7
  • A numerical method for solving three-dimensional, time-dependent incompressible Navier-Stokes equations in curvilinear coordinates is presented. The non-staggered-grid method originally developed by C. M. Rhie and W. L. Chow for steady state problems is extended to compute unsteady flows. In the computational space, the Cartesian velocity components and the pressure are defined at the center of a control volume, while the volume fluxes are defined at the mid-point on their corresponding cell faces. The momentum equations are integrated semi-implicitly by the approximate factorization technique. The intermediate velocities are interpolated onto the faces of the control volume to form the source termsmore » of the pressure Poisson equation, which is solved iteratively with a multigrid method. The compatibility condition of the pressure Poisson equation is satisfied in the same manner as in a staggered-grid method; mass conservation can be satisfied to machine accuracy. The pressure boundary condition is derived from the momentum equations. Solutions of both steady and unsteady problems including the large eddy simulation of a rotating and stratified upwelling flow in an irregular container established the favorable accuracy and efficiency of the present method.« less