### Abstract

The effects of discretization on the equations, and their solutions, describing Rayleigh–Bénard convection are studied through linear stability analysis and numerical integration of the discretized equations. Linear stability analyses of the discretized equations were conducted in the usual manner except that the assumed solution contained discretized components (e.g., spatial grid interval in the x direction, (Formula presented.). As the resolution became infinitely high (Formula presented.), the solutions approached those obtained from the continuous equations. The wavenumber of the maximum growth rate increased with increasing $${\Delta x}$$Δx until the wavenumber reached a minimum resolvable resolution, (Formula presented.). Therefore, the discretization of equations tends to reproduce higher-wavenumber structures than those predicted by the continuous equations. This behavior is counter intuitive and opposed to the expectation of (Formula presented.) leading to blurred simulated convection structures. However, when the analysis is conducted for discretized equations that are not combined into a single equation, as is the case for practically solved numerical models, the maximum growing wavenumber rather tends to decrease with increasing $${\Delta x}$$Δx as intuitively expected. The degree of the decrease depends on the discretization accuracy of the first-order differentials. When the accuracy of the discretization scheme is of low order, the wavenumber monotonically decreases with increasing (Formula presented.). On the other hand, when higher-order schemes are used for the discretization, the wavenumber does increase with increasing (Formula presented.), a similar trend to that in the case of the single-discretized equation for smaller (Formula presented.).

Original language | English |
---|---|

Pages (from-to) | 155-169 |

Number of pages | 15 |

Journal | Theoretical and Computational Fluid Dynamics |

Volume | 29 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2015 Jun 1 |

Externally published | Yes |

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### Keywords

- Convection
- Discretization error
- Linear stability analysis

### ASJC Scopus subject areas

- Computational Mechanics
- Condensed Matter Physics
- Engineering(all)
- Fluid Flow and Transfer Processes

### Cite this

*Theoretical and Computational Fluid Dynamics*,

*29*(3), 155-169. https://doi.org/10.1007/s00162-015-0345-x

**A linear thermal stability analysis of discretized fluid equations.** / Miyamoto, Yoshiaki; Ito, Junshi; Nishizawa, Seiya; Tomita, Hirofumi.

Research output: Contribution to journal › Article

*Theoretical and Computational Fluid Dynamics*, vol. 29, no. 3, pp. 155-169. https://doi.org/10.1007/s00162-015-0345-x

}

TY - JOUR

T1 - A linear thermal stability analysis of discretized fluid equations

AU - Miyamoto, Yoshiaki

AU - Ito, Junshi

AU - Nishizawa, Seiya

AU - Tomita, Hirofumi

PY - 2015/6/1

Y1 - 2015/6/1

N2 - The effects of discretization on the equations, and their solutions, describing Rayleigh–Bénard convection are studied through linear stability analysis and numerical integration of the discretized equations. Linear stability analyses of the discretized equations were conducted in the usual manner except that the assumed solution contained discretized components (e.g., spatial grid interval in the x direction, (Formula presented.). As the resolution became infinitely high (Formula presented.), the solutions approached those obtained from the continuous equations. The wavenumber of the maximum growth rate increased with increasing $${\Delta x}$$Δx until the wavenumber reached a minimum resolvable resolution, (Formula presented.). Therefore, the discretization of equations tends to reproduce higher-wavenumber structures than those predicted by the continuous equations. This behavior is counter intuitive and opposed to the expectation of (Formula presented.) leading to blurred simulated convection structures. However, when the analysis is conducted for discretized equations that are not combined into a single equation, as is the case for practically solved numerical models, the maximum growing wavenumber rather tends to decrease with increasing $${\Delta x}$$Δx as intuitively expected. The degree of the decrease depends on the discretization accuracy of the first-order differentials. When the accuracy of the discretization scheme is of low order, the wavenumber monotonically decreases with increasing (Formula presented.). On the other hand, when higher-order schemes are used for the discretization, the wavenumber does increase with increasing (Formula presented.), a similar trend to that in the case of the single-discretized equation for smaller (Formula presented.).

AB - The effects of discretization on the equations, and their solutions, describing Rayleigh–Bénard convection are studied through linear stability analysis and numerical integration of the discretized equations. Linear stability analyses of the discretized equations were conducted in the usual manner except that the assumed solution contained discretized components (e.g., spatial grid interval in the x direction, (Formula presented.). As the resolution became infinitely high (Formula presented.), the solutions approached those obtained from the continuous equations. The wavenumber of the maximum growth rate increased with increasing $${\Delta x}$$Δx until the wavenumber reached a minimum resolvable resolution, (Formula presented.). Therefore, the discretization of equations tends to reproduce higher-wavenumber structures than those predicted by the continuous equations. This behavior is counter intuitive and opposed to the expectation of (Formula presented.) leading to blurred simulated convection structures. However, when the analysis is conducted for discretized equations that are not combined into a single equation, as is the case for practically solved numerical models, the maximum growing wavenumber rather tends to decrease with increasing $${\Delta x}$$Δx as intuitively expected. The degree of the decrease depends on the discretization accuracy of the first-order differentials. When the accuracy of the discretization scheme is of low order, the wavenumber monotonically decreases with increasing (Formula presented.). On the other hand, when higher-order schemes are used for the discretization, the wavenumber does increase with increasing (Formula presented.), a similar trend to that in the case of the single-discretized equation for smaller (Formula presented.).

KW - Convection

KW - Discretization error

KW - Linear stability analysis

UR - http://www.scopus.com/inward/record.url?scp=84939971162&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939971162&partnerID=8YFLogxK

U2 - 10.1007/s00162-015-0345-x

DO - 10.1007/s00162-015-0345-x

M3 - Article

AN - SCOPUS:84939971162

VL - 29

SP - 155

EP - 169

JO - Theoretical and Computational Fluid Dynamics

JF - Theoretical and Computational Fluid Dynamics

SN - 0935-4964

IS - 3

ER -