A linear thermal stability analysis of discretized fluid equations

Yoshiaki Miyamoto, Junshi Ito, Seiya Nishizawa, Hirofumi Tomita

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)155-169
Number of pages15
JournalTheoretical and Computational Fluid Dynamics
Volume29
Issue number3
DOIs
Publication statusPublished - 2015 Jun 1
Externally publishedYes

Fingerprint

Thermodynamic stability
thermal stability
Fluids
fluids
Linear stability analysis
Numerical models
convection
numerical integration
Convection
counters
grids
intervals
trends
Direction compound

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

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

In: Theoretical and Computational Fluid Dynamics, Vol. 29, No. 3, 01.06.2015, p. 155-169.

Research output: Contribution to journalArticle

Miyamoto, Yoshiaki ; Ito, Junshi ; Nishizawa, Seiya ; Tomita, Hirofumi. / A linear thermal stability analysis of discretized fluid equations. In: Theoretical and Computational Fluid Dynamics. 2015 ; Vol. 29, No. 3. pp. 155-169.
@article{e8642a160f454646bfa847f5141ee20c,
title = "A linear thermal stability analysis of discretized fluid equations",
abstract = "The effects of discretization on the equations, and their solutions, describing Rayleigh–B{\'e}nard 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.).",
keywords = "Convection, Discretization error, Linear stability analysis",
author = "Yoshiaki Miyamoto and Junshi Ito and Seiya Nishizawa and Hirofumi Tomita",
year = "2015",
month = "6",
day = "1",
doi = "10.1007/s00162-015-0345-x",
language = "English",
volume = "29",
pages = "155--169",
journal = "Theoretical and Computational Fluid Dynamics",
issn = "0935-4964",
publisher = "Springer New York",
number = "3",

}

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 -