A roe-type numerical solver for the two-phase two-fluid six-equation model with realistic equation of state

Guojun Hu, Tomasz Kozlowski

Research output: Contribution to journalArticle

Abstract

In the nuclear industry, a method based on a staggered grid is used in two-phase flow system codes such as RELAP, TRAC, and CATHARE. Solving the two-phase two-fluid model with this method is complicated. The objective of this article is to develop a new solver, which is mathematically consistent and algebraically simpler than existing codes. The extension of existing shock-capturing upwind schemes for single-phase flows is our way. A numerical solver with a Roe-type numerical flux is formulated based on a very well-structured Jacobian matrix. We formulate the Jacobian matrix with arbitrary equation of state and simplify the Jacobian matrix to a simple and structured form with the help of a few auxiliary variables, e.g. isentropic speed of sound. Because the Jacobian matrix is very structured, the characteristic polynomial of the Jacobian matrix is simple and suitable for analytical analysis. Results from the characteristic analysis of the two-phase system are consistent with well-known facts, such as the ill-posedness of the basic two-phase two-fluid model which assumes all pressure terms are equal. An explicit numerical solver, with a Roe-type numerical flux, is constructed based on the characteristic analysis. A critical feature of the method is that the formulation does not depend on the form of equation of state and the method is applicable to realistic two-phase problems. We demonstrate solver performance based on three two-phase benchmark problems: two-phase shock-tube problem, faucet flow problem, and Christensen boiling pipe problem. The solutions are in excellent agreement with analytical solutions and numerical solutions from a system code. The new solver provides essential framework for developing a more accurate and robust solver for realistic reactor safety analysis. However, improvements on the new solver is necessary for achieving a high-order accuracy and increasing the robustness.

Original languageEnglish (US)
Pages (from-to)354-370
Number of pages17
JournalNuclear Engineering and Design
Volume326
DOIs
StatePublished - Jan 2018

Fingerprint

Jacobian matrices
Equations of state
equation of state
equations of state
matrix
Fluids
fluid
fluids
two fluid models
reactor safety
Fluxes
single-phase flow
binary systems (materials)
Nuclear industry
Shock tubes
Acoustic wave velocity
shock tubes
two phase flow
finite difference method
Two phase flow

Keywords

  • Hyperbolicity
  • Realistic EOS
  • Riemann solver
  • Two-phase flow

ASJC Scopus subject areas

  • Nuclear and High Energy Physics
  • Nuclear Energy and Engineering
  • Materials Science(all)
  • Safety, Risk, Reliability and Quality
  • Waste Management and Disposal
  • Mechanical Engineering

Cite this

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title = "A roe-type numerical solver for the two-phase two-fluid six-equation model with realistic equation of state",
abstract = "In the nuclear industry, a method based on a staggered grid is used in two-phase flow system codes such as RELAP, TRAC, and CATHARE. Solving the two-phase two-fluid model with this method is complicated. The objective of this article is to develop a new solver, which is mathematically consistent and algebraically simpler than existing codes. The extension of existing shock-capturing upwind schemes for single-phase flows is our way. A numerical solver with a Roe-type numerical flux is formulated based on a very well-structured Jacobian matrix. We formulate the Jacobian matrix with arbitrary equation of state and simplify the Jacobian matrix to a simple and structured form with the help of a few auxiliary variables, e.g. isentropic speed of sound. Because the Jacobian matrix is very structured, the characteristic polynomial of the Jacobian matrix is simple and suitable for analytical analysis. Results from the characteristic analysis of the two-phase system are consistent with well-known facts, such as the ill-posedness of the basic two-phase two-fluid model which assumes all pressure terms are equal. An explicit numerical solver, with a Roe-type numerical flux, is constructed based on the characteristic analysis. A critical feature of the method is that the formulation does not depend on the form of equation of state and the method is applicable to realistic two-phase problems. We demonstrate solver performance based on three two-phase benchmark problems: two-phase shock-tube problem, faucet flow problem, and Christensen boiling pipe problem. The solutions are in excellent agreement with analytical solutions and numerical solutions from a system code. The new solver provides essential framework for developing a more accurate and robust solver for realistic reactor safety analysis. However, improvements on the new solver is necessary for achieving a high-order accuracy and increasing the robustness.",
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author = "Guojun Hu and Tomasz Kozlowski",
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AU - Hu, Guojun

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N2 - In the nuclear industry, a method based on a staggered grid is used in two-phase flow system codes such as RELAP, TRAC, and CATHARE. Solving the two-phase two-fluid model with this method is complicated. The objective of this article is to develop a new solver, which is mathematically consistent and algebraically simpler than existing codes. The extension of existing shock-capturing upwind schemes for single-phase flows is our way. A numerical solver with a Roe-type numerical flux is formulated based on a very well-structured Jacobian matrix. We formulate the Jacobian matrix with arbitrary equation of state and simplify the Jacobian matrix to a simple and structured form with the help of a few auxiliary variables, e.g. isentropic speed of sound. Because the Jacobian matrix is very structured, the characteristic polynomial of the Jacobian matrix is simple and suitable for analytical analysis. Results from the characteristic analysis of the two-phase system are consistent with well-known facts, such as the ill-posedness of the basic two-phase two-fluid model which assumes all pressure terms are equal. An explicit numerical solver, with a Roe-type numerical flux, is constructed based on the characteristic analysis. A critical feature of the method is that the formulation does not depend on the form of equation of state and the method is applicable to realistic two-phase problems. We demonstrate solver performance based on three two-phase benchmark problems: two-phase shock-tube problem, faucet flow problem, and Christensen boiling pipe problem. The solutions are in excellent agreement with analytical solutions and numerical solutions from a system code. The new solver provides essential framework for developing a more accurate and robust solver for realistic reactor safety analysis. However, improvements on the new solver is necessary for achieving a high-order accuracy and increasing the robustness.

AB - In the nuclear industry, a method based on a staggered grid is used in two-phase flow system codes such as RELAP, TRAC, and CATHARE. Solving the two-phase two-fluid model with this method is complicated. The objective of this article is to develop a new solver, which is mathematically consistent and algebraically simpler than existing codes. The extension of existing shock-capturing upwind schemes for single-phase flows is our way. A numerical solver with a Roe-type numerical flux is formulated based on a very well-structured Jacobian matrix. We formulate the Jacobian matrix with arbitrary equation of state and simplify the Jacobian matrix to a simple and structured form with the help of a few auxiliary variables, e.g. isentropic speed of sound. Because the Jacobian matrix is very structured, the characteristic polynomial of the Jacobian matrix is simple and suitable for analytical analysis. Results from the characteristic analysis of the two-phase system are consistent with well-known facts, such as the ill-posedness of the basic two-phase two-fluid model which assumes all pressure terms are equal. An explicit numerical solver, with a Roe-type numerical flux, is constructed based on the characteristic analysis. A critical feature of the method is that the formulation does not depend on the form of equation of state and the method is applicable to realistic two-phase problems. We demonstrate solver performance based on three two-phase benchmark problems: two-phase shock-tube problem, faucet flow problem, and Christensen boiling pipe problem. The solutions are in excellent agreement with analytical solutions and numerical solutions from a system code. The new solver provides essential framework for developing a more accurate and robust solver for realistic reactor safety analysis. However, improvements on the new solver is necessary for achieving a high-order accuracy and increasing the robustness.

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