On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes

B. Thornber, D. Drikakis, R.J.R. Williams, D. Youngs

Research output: Contribution to journalArticle

67 Citations (Scopus)

Abstract

This paper addresses entropy generation and the corresponding dissipation of kinetic energy associated with high-resolution, shock-capturing (Godunov) methods. Analytical formulae are derived for the rate of increase of entropy given arbitrary jumps in primitive variables at a cell interface. It is demonstrated that for general continuously varying flows the inherent numerical entropy increase of Godunov methods is not proportional to the velocity jump cubed as is commonly assumed, but it is proportional to the velocity jump squared. Furthermore, the dissipation of kinetic energy is directly linked to temperature multiplied by change in entropy at low Mach numbers. The kinetic energy dissipation rate is shown to be proportional to the velocity jump squared and the speed of sound. The leading order dissipation rate associated with jumps in pressure, density and shear waves is detailed and further shown that at low Mach number it is the dissipation due to the perpendicular velocity jumps which dominates. This explains directly the poor performance of Godunov methods at low Mach numbers. The analysis is also applied to high-order accurate methods in space and time and all analytical results are validated with simple numerical experiments.
LanguageEnglish
Pages4853-4872
Number of pages20
JournalJournal of Computational Physics
Volume227
Issue number10
DOIs
Publication statusPublished - 1 May 2008

Fingerprint

Kinetic energy
Godunov method
Entropy
dissipation
kinetic energy
shock
entropy
Mach number
high resolution
Shear waves
Acoustic wave velocity
Energy dissipation
elastic waves
S waves
energy dissipation
acoustics
cells
Experiments
Temperature
temperature

Keywords

  • high-resolution methods
  • Godunov methods
  • dissipation
  • kinetic energy
  • entropy
  • large eddy simulation
  • low Mach number

Cite this

Thornber, B. ; Drikakis, D. ; Williams, R.J.R. ; Youngs, D. / On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes. In: Journal of Computational Physics. 2008 ; Vol. 227, No. 10. pp. 4853-4872.
@article{1a0f7528df374c6cb50c52d138a0c80f,
title = "On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes",
abstract = "This paper addresses entropy generation and the corresponding dissipation of kinetic energy associated with high-resolution, shock-capturing (Godunov) methods. Analytical formulae are derived for the rate of increase of entropy given arbitrary jumps in primitive variables at a cell interface. It is demonstrated that for general continuously varying flows the inherent numerical entropy increase of Godunov methods is not proportional to the velocity jump cubed as is commonly assumed, but it is proportional to the velocity jump squared. Furthermore, the dissipation of kinetic energy is directly linked to temperature multiplied by change in entropy at low Mach numbers. The kinetic energy dissipation rate is shown to be proportional to the velocity jump squared and the speed of sound. The leading order dissipation rate associated with jumps in pressure, density and shear waves is detailed and further shown that at low Mach number it is the dissipation due to the perpendicular velocity jumps which dominates. This explains directly the poor performance of Godunov methods at low Mach numbers. The analysis is also applied to high-order accurate methods in space and time and all analytical results are validated with simple numerical experiments.",
keywords = "high-resolution methods, Godunov methods, dissipation, kinetic energy, entropy, large eddy simulation, low Mach number",
author = "B. Thornber and D. Drikakis and R.J.R. Williams and D. Youngs",
year = "2008",
month = "5",
day = "1",
doi = "10.1016/j.jcp.2008.01.035",
language = "English",
volume = "227",
pages = "4853--4872",
journal = "Journal of Computational Physics",
issn = "0021-9991",
number = "10",

}

On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes. / Thornber, B.; Drikakis, D.; Williams, R.J.R.; Youngs, D.

In: Journal of Computational Physics, Vol. 227, No. 10, 01.05.2008, p. 4853-4872.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes

AU - Thornber, B.

AU - Drikakis, D.

AU - Williams, R.J.R.

AU - Youngs, D.

PY - 2008/5/1

Y1 - 2008/5/1

N2 - This paper addresses entropy generation and the corresponding dissipation of kinetic energy associated with high-resolution, shock-capturing (Godunov) methods. Analytical formulae are derived for the rate of increase of entropy given arbitrary jumps in primitive variables at a cell interface. It is demonstrated that for general continuously varying flows the inherent numerical entropy increase of Godunov methods is not proportional to the velocity jump cubed as is commonly assumed, but it is proportional to the velocity jump squared. Furthermore, the dissipation of kinetic energy is directly linked to temperature multiplied by change in entropy at low Mach numbers. The kinetic energy dissipation rate is shown to be proportional to the velocity jump squared and the speed of sound. The leading order dissipation rate associated with jumps in pressure, density and shear waves is detailed and further shown that at low Mach number it is the dissipation due to the perpendicular velocity jumps which dominates. This explains directly the poor performance of Godunov methods at low Mach numbers. The analysis is also applied to high-order accurate methods in space and time and all analytical results are validated with simple numerical experiments.

AB - This paper addresses entropy generation and the corresponding dissipation of kinetic energy associated with high-resolution, shock-capturing (Godunov) methods. Analytical formulae are derived for the rate of increase of entropy given arbitrary jumps in primitive variables at a cell interface. It is demonstrated that for general continuously varying flows the inherent numerical entropy increase of Godunov methods is not proportional to the velocity jump cubed as is commonly assumed, but it is proportional to the velocity jump squared. Furthermore, the dissipation of kinetic energy is directly linked to temperature multiplied by change in entropy at low Mach numbers. The kinetic energy dissipation rate is shown to be proportional to the velocity jump squared and the speed of sound. The leading order dissipation rate associated with jumps in pressure, density and shear waves is detailed and further shown that at low Mach number it is the dissipation due to the perpendicular velocity jumps which dominates. This explains directly the poor performance of Godunov methods at low Mach numbers. The analysis is also applied to high-order accurate methods in space and time and all analytical results are validated with simple numerical experiments.

KW - high-resolution methods

KW - Godunov methods

KW - dissipation

KW - kinetic energy

KW - entropy

KW - large eddy simulation

KW - low Mach number

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-41249085099&partnerID=40&md5=5f736c64bdea608d02d675d1b9e34642

U2 - 10.1016/j.jcp.2008.01.035

DO - 10.1016/j.jcp.2008.01.035

M3 - Article

VL - 227

SP - 4853

EP - 4872

JO - Journal of Computational Physics

T2 - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 10

ER -