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Evaluation of an effective and robust implicit time-integration numerical scheme for Navier-Stokes equations in a CFD solver for compressible flows

Author

Listed:
  • Maia, A.A.G.
  • Cavalca, D.F.
  • Tomita, J.T.
  • Costa, F.P.
  • Bringhenti, C.

Abstract

The present work describes the implementation of an implicit time-integration numerical scheme to solve viscous flows in an in-house CFD solver. The scheme is developed to calculate engineering problems involving compressible flows. This work extends the defect-correction technique for the 3D flow calculations, and all mathematical formulations are described. The CFD solver is based on the finite-volume method (FVM) to calculate the three-dimensional flow and can be applied to solve unstructured meshes. The current implementation uses the Flux-Difference Splitting method (FDS) developed by Roe combined with the MUSCL method and the Venkatakrishnan flux limiters to provide better accuracy of the numerical solutions. The implicit time-integration scheme was linearized applying the backward Euler method on the left-hand side (LHS) and a Newton-type linearization on the right-hand side (RHS) of the governing equations. The Jacobian matrix was computed analytically for the inviscid fluxes using the Roe fluxes, and for the viscous fluxes differentiating the conservative vector. Earlier work by Cavalca et al. (2018) showed the robustness and accuracy of this implicit solver to predict inviscid flows over the airfoil and into the supersonic nozzle. Finally, the Gauss-Seidel (GS) iterative method was applied to solve the resultant sparse and large system of equations. These numerical schemes and methods were applied to solve the laminar flow over a flat plate. Afterwards, the numerical solution was validated and verified with the exact Blasius solution. From the results, the numerical simulations exhibited superior robustness of the implicit-defect correction scheme when compared with the explicit scheme for compressible flows. All numerical particularities and their implementations are detailed in this paper.

Suggested Citation

  • Maia, A.A.G. & Cavalca, D.F. & Tomita, J.T. & Costa, F.P. & Bringhenti, C., 2022. "Evaluation of an effective and robust implicit time-integration numerical scheme for Navier-Stokes equations in a CFD solver for compressible flows," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    • Handle: RePEc:eee:apmaco:v:413:y:2022:i:c:s0096300321006962
    DOI: 10.1016/j.amc.2021.126612
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    References listed on IDEAS

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    1. Rubino, Samuele, 2019. "An efficient time-splitting approximation of the Navier–Stokes equations with LPS modeling," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 318-337.
    2. Huang, Zheng-Ge & Wang, Li-Gong & Xu, Zhong & Cui, Jing-Jing, 2020. "An efficient preconditioned variant of the PSS preconditioner for generalized saddle point problems," Applied Mathematics and Computation, Elsevier, vol. 376(C).
    3. Vorozhtsov, Evgenii V. & Shapeev, Vasily P., 2019. "On the efficiency of combining different methods for acceleration of iterations at the solution of PDEs by the method of collocations and least residuals," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    4. Zhao, Jingjun & Zhao, Wenjiao & Xu, Yang, 2021. "Lagrange nodal discontinuous Galerkin method for fractional Navier-Stokes equations," Applied Mathematics and Computation, Elsevier, vol. 391(C).
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    1. Cheng, Yong & Song, Fukai & Fu, Lei & Dai, Saishuai & Zhiming Yuan, & Incecik, Atilla, 2024. "Experimental investigation of a dual-pontoon WEC-type breakwater with a hydraulic-pneumatic complementary power take-off system," Energy, Elsevier, vol. 286(C).

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