A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh

Abstract

High-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.

Preconditioned pseudo-compressibility methods for Three-dimensional incompressible Navier-Stokes equations

“…According to their numerical tests, Malan et al [14,15] proposed that when α≤1 this preconditioned pseudo-compressibility procedure presents strong robustness, and they also proposed an adaptive α. Reference [13] suggested α=1 to balance the convergence property and the robustness. However, all the above mentioned works have not explored the exact reason of the non-robustness of the present algorithm, especially since central schemes are adopted in [14,15], which do not involve the eigen-systems of the Jacbian matrix.…”

“…Since the Jacobian matrix contains two free parameters α and β, the unified form of the eigenvalues and eigenvectors of the Jacobian matrix are hard to be obtained in the three-dimensional case. Recently, some work involved the eigen-systems of the three-dimensional case, Charest et al [13] proposed the eigenvalues and eigenvectors when α=1 and established the high order CENO scheme for the three-dimensional flow.…”

A cartesian grid embedded boundary method for the compressible Navier–Stokes equations

“…See Schlichting [26] or White [32] for a full exposition of this derivation. Charest et al [6] present a low-Mach-number algorithm for steady state calculations. In this calculation, they present a boundary layer calculation that reproduces the behavior Blasius layer.…”

“…Charest et al [6] present a low-Machnumber algorithm for steady state calculations. They present a boundary layer calculation that reproduces the behavior of the similarity solution which emerges from analysis (a Blasius boundary layer profile).…”

A High-Order Finite-Volume Method with Anisotropic AMR for Ideal MHD Flows

“…4 The efficiency of the CENO scheme has also been assessed on cubed-sphere meshes. 4 Moreover the method has been extended to unstructured meshes for laminar viscous flows 7 and turbulent reactive flows. 8 Block-based adaptive mesh refinement (AMR) approaches [9][10][11] are very attractive since they allow to refine automatically the mesh for treating the disparate spatial scales while requiring an overall light data structure due to grid connectivity being computed on a block level.…”