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Numerical analysis of a 1/2-equation model of turbulence

Wei-Wei Han, Rui Fang, William Layton

TL;DR

The paper provides a rigorous numerical analysis of the 1/2-equation turbulence model, a space-averaged simplification of the 1-equation URANS framework, proving uniqueness of strong solutions and establishing stability, convergence, and error estimates for both semi-discrete and fully discrete finite-element schemes. By handling the nonmonotone eddy-viscosity nonlinearity and the cubic right-hand side of the k-equation, the authors derive error bounds that separate initial and approximation errors, and they confirm the expected temporal and spatial convergence rates through numerical tests. The results indicate that the 1/2-equation model can capture key velocity statistics at reduced computational cost compared to fully resolved models, providing a solid numerical foundation for its use and further study toward the full 1-equation model.

Abstract

The recent 1/2-equation model of turbulence is a simplification of the standard Kolmogorov-Prandtl 1-equation URANS model. Surprisingly, initial numerical tests indicated that the 1/2-equation model produces comparable velocity statistics at reduced cost. It is also a test problem and first step for developing numerical analysis to address a full 1-equation model. This report begins the numerical analysis of the 1/2 equation model. Stability, convergence and error estimates are proven for a semi-discrete and fully discrete approximation. Finally, numerical tests are conducted to validate our convergence theory.

Numerical analysis of a 1/2-equation model of turbulence

TL;DR

The paper provides a rigorous numerical analysis of the 1/2-equation turbulence model, a space-averaged simplification of the 1-equation URANS framework, proving uniqueness of strong solutions and establishing stability, convergence, and error estimates for both semi-discrete and fully discrete finite-element schemes. By handling the nonmonotone eddy-viscosity nonlinearity and the cubic right-hand side of the k-equation, the authors derive error bounds that separate initial and approximation errors, and they confirm the expected temporal and spatial convergence rates through numerical tests. The results indicate that the 1/2-equation model can capture key velocity statistics at reduced computational cost compared to fully resolved models, providing a solid numerical foundation for its use and further study toward the full 1-equation model.

Abstract

The recent 1/2-equation model of turbulence is a simplification of the standard Kolmogorov-Prandtl 1-equation URANS model. Surprisingly, initial numerical tests indicated that the 1/2-equation model produces comparable velocity statistics at reduced cost. It is also a test problem and first step for developing numerical analysis to address a full 1-equation model. This report begins the numerical analysis of the 1/2 equation model. Stability, convergence and error estimates are proven for a semi-discrete and fully discrete approximation. Finally, numerical tests are conducted to validate our convergence theory.
Paper Structure (9 sections, 7 theorems, 84 equations, 2 tables)

This paper contains 9 sections, 7 theorems, 84 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Model derivation
  3. Related work
  4. Preliminaries
  5. Model Uniqueness
  6. The semi-discrete approximation
  7. The fully discrete approximation
  8. Numerical Tests
  9. Conclusions

Key Result

Lemma 2.1

Suppose that $G,\Delta t$, and $d_n, e_n, a_n, b_n$ (for integer $n\geq0$) be nonnegative numbers such that for $\forall N \geq 1$ and $\forall \Delta t > 0$, then

Theorems & Definitions (12)

  • Lemma 2.1
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • proof
  • ...and 2 more