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Identifying the body force from partial observations of a 2D incompressible velocity field

Aseel Farhat, Adam Larios, Vincent R. Martinez, Jared P. Whitehead

Abstract

Using limited observations of the velocity field of the two-dimensional Navier-Stokes equations, we successfully reconstruct the steady body force that drives the flow. The number of observed data points is less than 10\% of the number of modes that describes the full flow field, indicating that the method introduced here is capable of identifying complicated forcing mechanisms from a relatively small collection of observations. In addition to demonstrating the efficacy of this method on turbulent flow data generated by simulations of the two-dimensional Navier-Stokes equations, we also rigorously justify convergence of the derived algorithm. Beyond the practical applicability of such an algorithm, the reliance of this method on the dynamical evolution of the system yields physical insight into the turbulent cascade.

Identifying the body force from partial observations of a 2D incompressible velocity field

Abstract

Using limited observations of the velocity field of the two-dimensional Navier-Stokes equations, we successfully reconstruct the steady body force that drives the flow. The number of observed data points is less than 10\% of the number of modes that describes the full flow field, indicating that the method introduced here is capable of identifying complicated forcing mechanisms from a relatively small collection of observations. In addition to demonstrating the efficacy of this method on turbulent flow data generated by simulations of the two-dimensional Navier-Stokes equations, we also rigorously justify convergence of the derived algorithm. Beyond the practical applicability of such an algorithm, the reliance of this method on the dynamical evolution of the system yields physical insight into the turbulent cascade.
Paper Structure (8 sections, 33 equations, 3 figures)

This paper contains 8 sections, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The equations of motion and their mathematical setting
  3. Derivation of the algorithm and Heuristics for convergence
  4. Definition of the algorithm
  5. Rigorous Convergence Analysis
  6. Simulation Results
  7. Conclusions
  8. Appendix

Figures (3)

  • Figure 2: Convergence plots for the exact Laplacian algorithm with force updated continuously (i.e., at each time step) in various norms. Comparing (a) and (b), it appears that observing all modes contained in the force gives convergence to machine precision, while not observing only a small number of modes leads to lack of convergence.
  • Figure 3: (log-linear plot) $L^2$ errors vs. time. "DR" refers to the direct replacement algorithm, while "EX" refers to the exact Laplacian algorithm. "-c" indicates that the force was updated continuously (i.e., at each time step), while "-d" indicates discrete force updates, every $0.25$ time units.
  • Figure 4: (log-log plot) Spectra of the error in the stream function (a) and the error of the forcing function (b) at times $t=0.0, 0.5, 1.0$, and also $t=2, 4, 6\ldots,40$ using direct replacement scheme. Colors move from blue ($t=0.0$) to red ($t=40.0)$. Vertical dashed line is the observational wave-number cut-off at $|\mathbf{k}|_\infty=64$. Horizontal dotted line is machine precision ($\epsilon\approx2.22\times10^{-16}$). (c) Relative error in spectrum spectrum at times $t=0.0, 0.5, 1.0$, and also $t=2, 4, 6\ldots,80$ using direct replacement scheme.(d) Relative error in forcing spectrum at times $t=0.0, 0.5, 1.0$, and also $t=2, 4, 6\ldots,80$ using direct replacement scheme.

Theorems & Definitions (3)

  • Remark 4.1
  • Remark 4.2
  • Remark 1