A High-Order Hybrid-Spectral Incompressible Navier-Stokes Model For Nonlinear Water Waves
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arXiv
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| Format: | Preprint |
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2024
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| author | Melander, Anders Bitsch, Max E. Chen, Dong Engsig-Karup, Allan P. |
| author_facet | Melander, Anders Bitsch, Max E. Chen, Dong Engsig-Karup, Allan P. |
| contents | We present a new high-order accurate computational fluid dynamics model based on the incompressible Navier-Stokes equations with a free surface for the accurate simulation of nonlinear and dispersive water waves in the time domain. The spatial discretization is based on Chebyshev polynomials in the vertical direction and a Fourier basis in the horizontal direction, allowing for the use of the fast Chebyshev and Fourier transforms for the efficient computation of spatial derivatives. The temporal discretization is done through a generalized low-storage explicit 4th order Runge-Kutta, and for the scheme to conserve mass and achieve high-order accuracy, a velocity-pressure coupling needs to be satisfied at all Runge-Kutta stages. This result in the emergence of a Poisson pressure problem that constitute a geometric conservation law for mass conservation. The occurring Poisson problem is proposed to be solved efficiently via an accelerated iterative solver based on a geometric $p$-multigrid scheme, which takes advantage of the high-order polynomial basis in the spatial discretization and hence distinguishes itself from conventional low-order numerical schemes. We present numerical experiments for validation of the scheme in the context of numerical wave tanks demonstrating that the $p$-multigrid accelerated numerical scheme can effectively solve the Poisson problem that constitute the computational bottleneck, that the model can achieve the desired spectral convergence, and is capable of simulating wave-propagation over non-flat bottoms with excellent agreement in comparison to experimental results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A High-Order Hybrid-Spectral Incompressible Navier-Stokes Model For Nonlinear Water Waves Melander, Anders Bitsch, Max E. Chen, Dong Engsig-Karup, Allan P. Numerical Analysis Fluid Dynamics We present a new high-order accurate computational fluid dynamics model based on the incompressible Navier-Stokes equations with a free surface for the accurate simulation of nonlinear and dispersive water waves in the time domain. The spatial discretization is based on Chebyshev polynomials in the vertical direction and a Fourier basis in the horizontal direction, allowing for the use of the fast Chebyshev and Fourier transforms for the efficient computation of spatial derivatives. The temporal discretization is done through a generalized low-storage explicit 4th order Runge-Kutta, and for the scheme to conserve mass and achieve high-order accuracy, a velocity-pressure coupling needs to be satisfied at all Runge-Kutta stages. This result in the emergence of a Poisson pressure problem that constitute a geometric conservation law for mass conservation. The occurring Poisson problem is proposed to be solved efficiently via an accelerated iterative solver based on a geometric $p$-multigrid scheme, which takes advantage of the high-order polynomial basis in the spatial discretization and hence distinguishes itself from conventional low-order numerical schemes. We present numerical experiments for validation of the scheme in the context of numerical wave tanks demonstrating that the $p$-multigrid accelerated numerical scheme can effectively solve the Poisson problem that constitute the computational bottleneck, that the model can achieve the desired spectral convergence, and is capable of simulating wave-propagation over non-flat bottoms with excellent agreement in comparison to experimental results. |
| title | A High-Order Hybrid-Spectral Incompressible Navier-Stokes Model For Nonlinear Water Waves |
| topic | Numerical Analysis Fluid Dynamics |
| url | https://arxiv.org/abs/2406.00991 |