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Temperature-driven turbulence in compressible fluid flows

Eduard Feireisl, Maria Lukacova-Medvidova, Bangwei She, Yuhuan Yuan

Abstract

We study the long-time behaviour of the temperature-driven compressible flows. We show that numerical solutions of a structure-preserving finite volume method generate a discrete attractor that consists of entire discrete trajectories. Further, we prove the convergence of discrete attractors to their continuous counterparts. Theoretical results are illustrated by extensive numerical simulations of the well-known Rayleigh-Benard problem. The numerical results also indicate the validity of the ergodic hypothesis and imply that a non-zero Reynolds stress persist for long time. Finally, we also observe that any invariant measure is of Gaussian type in sharp contrast with the conjecture proposed by [Glimm et al., SN Applied Sciences 2, 2160 (2020)].

Temperature-driven turbulence in compressible fluid flows

Abstract

We study the long-time behaviour of the temperature-driven compressible flows. We show that numerical solutions of a structure-preserving finite volume method generate a discrete attractor that consists of entire discrete trajectories. Further, we prove the convergence of discrete attractors to their continuous counterparts. Theoretical results are illustrated by extensive numerical simulations of the well-known Rayleigh-Benard problem. The numerical results also indicate the validity of the ergodic hypothesis and imply that a non-zero Reynolds stress persist for long time. Finally, we also observe that any invariant measure is of Gaussian type in sharp contrast with the conjecture proposed by [Glimm et al., SN Applied Sciences 2, 2160 (2020)].

Paper Structure

This paper contains 22 sections, 8 theorems, 126 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Available analytical results
  3. Objectives addressed in the present paper
  4. Organization of the paper
  5. Numerical method
  6. Notation
  7. Scheme
  8. Time interpolation
  9. Hypothesis
  10. Discrete attractors
  11. Convergence of discrete attractors
  12. Stability and consistency
  13. Proof of Theorem \ref{['CT1']}
  14. Step 1
  15. Step 2
  16. ...and 7 more sections

Key Result

Theorem 3.3

Let $\{\widetilde{{\bf U}}_h^n\}_{n=1}^\infty$ be a sequence of discrete solutions defined on the time intervals $[T_n, \infty)$, $T_n \to - \infty$. Suppose that the associated numerical solutions $\widetilde{{\bf U}}_h^n$ satisfy Hypothesis (B) (with the same bounds $\underline{\varrho}$, $\overli where $\widetilde{{\bf U}}_h^\infty$ is an entire discrete solution.

Figures (12)

  • Figure 2: Rayleigh--Bénard Experiment 1: $\left\lVert U_h(T_M,\cdot)\right\rVert_{L^1(\Omega)}$ (top) and $\overline{\left\lVert U_h(T_M,\cdot)\right\rVert_{L^1(\Omega)}}$ (bottom).
  • Figure 3: Rayleigh--Bénard Experiment 1: Numerical solutions $(\varrho_h, u_{1,h}, u_{2,h}, \vartheta_h)$ (from left to right, from top to bottom) at $T = 800$.
  • Figure 4: Rayleigh--Bénard Experiment 1: Errors $\widetilde{E_1}, \widetilde{E_2}$ over $[0,800]$.
  • Figure 16: Rayleigh--Bénard Experiments: evolutions of $\left\lVert m_{1,h}(t,\cdot)\right\rVert_{L^1(\Omega)}$ (top) and $\left\lVert E_h(t,\cdot)\right\rVert_{L^1(\Omega)}$ (bottom) for Experiments 2-5 (from left to right).
  • Figure 17: Rayleigh--Bénard Experiments: errors $E_i, \, i = 1,2,3,4$ (from top to bottom) for Experiments 2-5 (from left to right).
  • ...and 7 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Weak form
  • Definition 2.4: Strong form
  • Definition 3.1: Discrete solution
  • Definition 3.2: Entire discrete solution
  • Theorem 3.3: Discrete attractor
  • proof
  • Theorem 4.1: Attractor convergence
  • Lemma 5.1: Uniform bounds FeLMShYu:2024
  • ...and 7 more