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A simple and efficient incompressible Navier-Stokes solver for unsteady complex geometry flows on truncated domains
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
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2017 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 150, p. 84-94Article in journal (Refereed) Published
Abstract [en]

Incompressible Navier-Stokes solvers based on the projection method often require an expensive numerical solution of a Poisson equation for a pressure-like variable. This often involves linear system solvers based on iterative and multigrid methods which may limit the ability to scale to large numbers of processors. The artificial compressibility method (ACM) [6] introduces a time derivative of the pressure into the incompressible form of the continuity equation creating a coupled closed hyperbolic system that does not require a Poisson equation solution and allows for explicit time-marching and localized stencil numerical methods. Such a scheme should theoretically scale well on large numbers of CPUs, GPU'€™s, or hybrid CPU-GPU architectures. The original ACM was only valid for steady flows and dual-time stepping was often used for time-accurate simulations. Recently, Clausen [7] has proposed the entropically damped artificial compressibility (EDAC) method which is applicable to both steady and unsteady flows without the need for dual-time stepping. The EDAC scheme was successfully tested with both a finite-difference MacCormack'€™s method for the two-dimensional lid driven cavity and periodic double shear layer problem and a finite-element method for flow over a square cylinder, with scaling studies on the latter to large numbers of processors. In this study, we discretize the EDAC formulation with a new optimized high-order centered finite-difference scheme and an explicit fourth-order Runge-€“Kutta method. This is combined with an immersed boundary method to efficiently treat complex geometries and a new robust outflow boundary condition to enable higher Reynolds number simulations on truncated domains. Validation studies for the Taylor-€“Green Vortex problem and the lid driven cavity problem in both 2D and 3D are presented. An eddy viscosity subgrid-scale model is used to enable large eddy simulations for the 3D cases. Finally, an application to flow over a sphere is presented to highlight the boundary condition and performance comparisons to a traditional incompressible Navier-€“Stokes solver is shown for the 3D lid driven cavity. Overall, the combined EDAC formulation and discretization is shown to be both effective and affordable.

Place, publisher, year, edition, pages
Elsevier, 2017. Vol. 150, p. 84-94
Keywords [en]
Artificial compressibility method, EDAC, High-order numerical methods, Large Eddy simulation
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-136507DOI: 10.1016/j.compfluid.2017.03.030ISI: 000401219000007OAI: oai:DiVA.org:liu-136507DiVA, id: diva2:1089226
Note

Funding agencies: Rosenblatt Chair within the faculty of Mechanical Engineering; Zeff Fellowship Trust

Available from: 2017-04-19 Created: 2017-04-19 Last updated: 2017-11-20Bibliographically approved
In thesis
1. Error analysis of summation-by-parts formulations: Dispersion, transmission and accuracy
Open this publication in new window or tab >>Error analysis of summation-by-parts formulations: Dispersion, transmission and accuracy
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximation-terms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems.

The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times.

In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid.

In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. p. 27
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1886
National Category
Computational Mathematics Mathematical Analysis Control Engineering Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:liu:diva-143059 (URN)10.3384/diss.diva-143059 (DOI)978-91-7685-427-3 (ISBN)
Public defence
2017-12-12, Ada Lovelace,, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2017-11-20 Created: 2017-11-20 Last updated: 2017-11-20Bibliographically approved

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The full text will be freely available from 2019-04-01 08:00
Available from 2019-04-01 08:00

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Nordström, JanLinders, Viktor

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