The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.@en

The modified augmented Lagrangian preconditioner has attracted much attention in solving nondimensional Navier-Stokes equations discretized by the finite element method. In industrial applications the governing equations are often in dimensional form and discretized using the finite volume method. This paper assesses the capability of this preconditioner for dimensional Navier-Stokes equations in the context of the finite volume method. Two main contributions are made. First, this paper introduces a new dimensionless parameter that is involved in the modified augmented Lagrangian preconditioner. Second, with a number of academic test problems this paper reveals that the convergence of both nonlinear and linear iterations depend on this dimensionless parameter. A way to choose the optimal value of the dimensionless parameter is suggested and it is found that the optimal value is dependent of the Reynolds number, instead of the fluid's properties, e.g., density and dynamic viscosity. The outcomes of this paper yield a potential rule to choose the optimal dimensionless parameter in practice, namely, correspondingly increasing with enlarging the Reynolds number.@en

In this paper we explore the performance of the SIMPLER, augmented Lagrangian, ‘grad-div’ preconditioners and their new variants for the two-by-two block systems arising in the incompressible Navier-Stokes equations. The lid-driven cavity and flow over a finite flat plate are chosen as the benchmark problems. For each problem the Reynolds number varies from a low to the limiting number for a laminar flow.@en

The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.@en

The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.@en

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