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Parameter Analysis in Continuous Data Assimilation for Various Turbulence Models

Debora A. F. Albanez, Maicon Jose Benvenutti, Samuel Little, Jing Tian

Abstract

In this study, we conduct parameter estimation analysis on a data assimilation algorithm for two turbulence models: the simplified Bardina model and the Navier-Stokes-α model. Our approach involves creating an approximate solution for the turbulence models by employing an interpolant operator based on the observational data of the systems. The estimation depends on the parameter alpha in the models. Additionally, numerical simulations are presented to validate our theoretical results

Parameter Analysis in Continuous Data Assimilation for Various Turbulence Models

Abstract

In this study, we conduct parameter estimation analysis on a data assimilation algorithm for two turbulence models: the simplified Bardina model and the Navier-Stokes-α model. Our approach involves creating an approximate solution for the turbulence models by employing an interpolant operator based on the observational data of the systems. The estimation depends on the parameter alpha in the models. Additionally, numerical simulations are presented to validate our theoretical results
Paper Structure (16 sections, 6 theorems, 112 equations, 24 figures)

This paper contains 16 sections, 6 theorems, 112 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical settings
  4. Abstract formulations
  5. Estimates for reference solutions
  6. Results on the long-time error
  7. Numerical Simulation
  8. Testing the impact of $\eta$-without random initial conditions
  9. Numerical Simulation for the simplified Bardina model
  10. Numerical Simulation for the Navier-Stokes-$\alpha$ model
  11. Testing the impact of $\eta$-with random initial conditions
  12. Numerical Simulation for the simplified Bardina model
  13. Numerical Simulation for the Navier-Stokes-$\alpha$ model
  14. Discussions on the numerical computations
  15. Conclusions and Discussions
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\xi:[t_0,\infty)\rightarrow[0,\infty)$ absolute continuous function and $\beta:[t_0,\infty)\rightarrow[0,\infty)$ locally integrable. Suppose that there exist positive constants $C,M$ and $T$ such that is satisfied for all $t\geq t_0$. Then for all $t\geq t_0$.

Figures (24)

  • Figure 1: Error plot of Bardina model with high $\eta$ value-without random initial conditions case.
  • Figure 2: Velocity contour of Bardina model with high $\eta$ value-without random initial conditions case at $t=0$.
  • Figure 3: Velocity contour of Bardina model with high $\eta$ value-without random initial conditions case at $t=400$.
  • Figure 4: Error plot of Bardina model with high $\eta$ value-without random initial conditions case.
  • Figure 5: Velocity contour of Bardina model with high $\eta$ value-without random initial conditions case at $t=0$.
  • ...and 19 more figures

Theorems & Definitions (12)

  • Lemma 1: Gronwall Inequality
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 2 more