Mesh-free Navier-Stokes and Stability

2024

1. Mesh-free hydrodynamic stability

   T. Chu, and O. T. Schmidt

   Journal of Computational Physics, 2024, 502, 112822

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   A specialized mesh-free radial basis function-based finite difference (RBF-FD) discretization is used to solve the large eigenvalue problems arising in hydrodynamic stability analyses of flows in complex domains. Polyharmonic spline functions with polynomial augmentation (PHS+poly) are used to construct the discrete linearized incompressible and compressible Navier-Stokes operators on scattered nodes. Rigorous global and local eigenvalue stability studies of these global operators and their constituent RBF stencils provide a set of parameters that guarantee stability without the need for hyperviscosity or other ad hoc regularizations, while balancing accuracy and computational efficiency. Specialized elliptical stencils to compute boundary-normal derivatives are introduced, and the treatment of reflectional and rotational symmetries is discussed. In particular, treating the pole singularity in cylindrical coordinates permits dimensionality reduction from 3D to 2D in rotational flows like jets. The numerical framework is demonstrated and validated on several hydrodynamic stability methods ranging from classical linear theory of laminar flows to state-of-the-art non-modal approaches that are applicable to turbulent mean flows. The examples include linear stability, resolvent, and wavemaker analyses of cylinder flow at Reynolds numbers ranging from 47 to 180, and resolvent and wavemaker analyses of the self-similar flat-plate boundary layer at a Reynolds number as well as the turbulent mean of a high-Reynolds-number transonic jet at Mach number 0.9. All previously-known results are found in close agreement with the literature. Finally, the resolvent-based wavemaker analyses of the Blasius boundary layer and turbulent jet flows offer new physical insight into the modal and non-modal growth in these flows.

2023

1. RBF-FD discretization of the Navier-Stokes equations on scattered but staggered nodes

   T. Chu, and O. T. Schmidt

   Journal of Computational Physics, 2023, 474, 111756

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   A semi-implicit fractional-step method that uses a staggered node layout and radial basis function-finite differences (RBF-FD) to solve the incompressible Navier-Stokes equations is developed. Polyharmonic splines (PHS) with polynomial augmentation (PHS+poly) are used to construct the global differentiation matrices. A systematic parameter study identifies a combination of stencil size, PHS exponent, and polynomial degree that minimizes the truncation error for a wave-like test function on scattered nodes. Classical modified wavenumber analysis is extended to RBF-FDs on heterogeneous node distributions and used to confirm that the accuracy of the selected 28-point stencil is comparable to that of spectral-like, 6th-order Padé-type finite differences. The Navier-Stokes solver is demonstrated on two benchmark problems, internal flow in a lid-driven cavity in the Reynolds number regime 102≤Re≤104, and open flow around a cylinder at Re=100 and 200. The combination of grid staggering and careful parameter selection facilitates accurate and stable simulations at significantly lower resolutions than previously reported, using more compact RBF-FD stencils, without special treatment near solid walls, and without the need for hyperviscosity or other means of regularization.

2. RBF-FD-based global resolvent analysis: the Blasius boundary layer

   T. Chu, and O. T. Schmidt

   AIAA Paper 2023-3567, 2023

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   View Video Presentation: https://doi.org/10.2514/6.2023-3567.vidA high-order mesh-free hydrodynamic stability analysis framework is developed based on radial basis function-based finite differences (RBF-FD). The global linearized Navier-Stokes operator is discretized using polyharmonic spline RBFs with polynomial augmentation (PHS+poly). As a validation test case, the non-parallel flat-plate boundary layer within a computational domain corresponding to a local Reynolds number of 0 ≤ Re_x ≤ 6x10^5 is considered. The computational domain is discretized using scattered nodes created by an unstructured mesh generator. The discrete resolvent operator is constructed, and its singular components are analyzed. The resulting optimal response modes are identified as Tollmien-Schlichting (TS) wavepackets. These responses are optimally forced by the upstream tilted structures that leverage the Orr mechanism. Favorable comparisons to previous literature, both qualitatively and quantitatively, validate the novel framework.

2022

1. Mesh-free RBF-based discretizations for hydrodynamic stability analysis

   T. Chu, and O. T. Schmidt

   AIAA Paper 2022-4098, 2022

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   View Video Presentation: https://doi.org/10.2514/6.2022-4098.vid Radial basis function-based finite differences (RBF-FD) are used to develop a high-order mesh-free hydrodynamic stability analysis tool for complex geometries. Polyharmonic spline RBFs with polynomial augmentation (PHS+poly) are used to construct the discrete linearized Navier-Stokes and resolvent operators on arbitrarily scattered nodes. The PHS+poly discretization is shown to yield accurate, stable, and computationally efficient discretizations of the large hydrodynamic stability matrix problems arising in two-dimensional classical linear theory and resolvent analysis. The mean-flow stability of the wake behind a cylinder in the vicinity of the critical point is studied using both theories. The predicted flow instabilities, including the vortex shedding frequency and associated coherent structures, closely match those reported in the literature.

2021

1. An RBF-based finite difference discretization of the Navier-Stokes equations: error analysis and application to lid-driven cavity flow

   T. Chu, and O. T. Schmidt

   AIAA Paper 2021-2743, 2021

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   View Video Presentation: https://doi.org/10.2514/6.2021-2743.vidRadial basis function-finite differences (RBF-FD) are used to solve the incompressible Navier-Stokes equations on scattered nodes. We present a semi-implicit fractional-step method that uses a staggered grid arrangement. The RBF-QR method devised by Fornberg and Piret[1] is used to obtain the RBF-FD weights for the spatial derivatives. We propose a rigorous error analysis strategy to identify optimal combinations of the shape parameter, ε, and the stencil size,n. A modified wavenumber analysis shows that the accuracy of the RBF differentiation matrices based on the optimal parameters is comparable to 4th-order Padé-type finite differences, for both first and second derivatives. The internal flow in a lid-driven cavity is studied as an example. We demonstrate that stable solutions are obtained without the need for hyperviscosity.