dorsal/arxiv
View SchemaGeneralized theory for numerical instability of the Gaussian-filtered Navier-Stokes equations as a model system for large eddy simulation of turbulence
| Authors | Masato Ida, Nobuyuki Oshima |
|---|---|
| Categories | |
| ArXiv ID | physics/0606196 |
| URL | https://arxiv.org/abs/physics/0606196 |
Abstract
The Gaussian-filtered Navier-Stokes equations are examined theoretically and a generalized theory of their numerical stability is proposed. Using the exact expansion series of subfilter-scale stresses or integration by parts, the terms describing the interaction between the mean and fluctuation portions in a statistically steady state are theoretically rewritten into a closed form in terms of the known filtered quantities. This process involves high-order derivatives with time-independent coefficients. Detailed stability analyses of the closed formulas are presented for determining whether a filtered system is numerically stable when finite difference schemes or others are used to solve it. It is shown that by the Gaussian filtering operation, second and higher even-order derivatives are derived that always exhibit numerical instability in a fixed range of directions; hence, if the filter widths are unsuitably large, the filtered Navier-Stokes equations can in certain cases be unconditionally unstable even though there is no error in modeling the subfilter-scale stress terms. As is proved by a simple example, the essence of the present discussion can be applied to any other smooth filters; that is, such a numerical instability problem can arise whenever the dependent variables are smoothed out by a filter.
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"abstract": "The Gaussian-filtered Navier-Stokes equations are examined theoretically and\na generalized theory of their numerical stability is proposed. Using the exact\nexpansion series of subfilter-scale stresses or integration by parts, the terms\ndescribing the interaction between the mean and fluctuation portions in a\nstatistically steady state are theoretically rewritten into a closed form in\nterms of the known filtered quantities. This process involves high-order\nderivatives with time-independent coefficients. Detailed stability analyses of\nthe closed formulas are presented for determining whether a filtered system is\nnumerically stable when finite difference schemes or others are used to solve\nit. It is shown that by the Gaussian filtering operation, second and higher\neven-order derivatives are derived that always exhibit numerical instability in\na fixed range of directions; hence, if the filter widths are unsuitably large,\nthe filtered Navier-Stokes equations can in certain cases be unconditionally\nunstable even though there is no error in modeling the subfilter-scale stress\nterms. As is proved by a simple example, the essence of the present discussion\ncan be applied to any other smooth filters; that is, such a numerical\ninstability problem can arise whenever the dependent variables are smoothed out\nby a filter.",
"arxiv_id": "physics/0606196",
"authors": [
"Masato Ida",
"Nobuyuki Oshima"
],
"categories": [
"physics.flu-dyn",
"physics.comp-ph"
],
"title": "Generalized theory for numerical instability of the Gaussian-filtered Navier-Stokes equations as a model system for large eddy simulation of turbulence",
"url": "https://arxiv.org/abs/physics/0606196"
},
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"variant": "snapshot-2026-03-01",
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