Continuous data assimilation for the 2D Bénard convection through velocity measurements alone

Abstract

Abstract. An algorithm for continuous data assimilation for the two-dimensional Bénard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier-Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate, the unique exact unknown solution of the Bénard convection problem associated with the observed (finite dimensional projection of) velocity.MSC Subject Classifications: 35Q30, 93C20, 37C50, 76B75, 34D06.

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

“…The analytical results reported in [15] were demonstrated numerically, for low Rayleigh numbers, in [2] where evidence was also provided to show that assimilation can fail when using temperature measurement data alone. The works in [2,15] were performed under no-slip boundary conditions for the velocity vector field at the horizontal solid walls. The question whether temperature observations are enough to determine all the dynamical state of the system is an important practical problem in meteorology and engineering and referred to as Charney's conjecture [11].…”

“…The Rayleigh-Bénard system governs two fields, velocity and temperature. It was proved in [15], that both velocity and temperature can be recovered using only velocity coarse-mesh spatial measurements. Later in [18], it was shown that a modification of the algorithm (2.2) for the Rayleigh-Bénard system, with no penetration stress-free boundary conditions on the horizontal walls, works by employing only coarse-mesh spatial measurements for the horizontal component of velocity vector field.…”

“…In this paper we carry out a computational experiment similar to that in [2], but at higher Rayleigh numbers and on the vorticity formulation of the Rayleigh-Bénard system. We propose, and computationally test, a data assimilation algorithm in the spirit of the algorithm in [4,15], that uses coarse-grain vorticity or local circulation spatial measurements only rather than velocity measurements, to recover the full state of the system. The algorithm is presented in section 3 and the numerical results are presented in section 4.…”

High–low frequency slaving and regularity issues in the 3D Navier–Stokes equations

“…Some success was achieved for the onedimensional the Kuramoto-Sivashinsky equation where an inertial manifold was proved to exist (Foias, Jolly, Kevrekidis, Sell and Titi 1988). Further success was also achieved for the 2D incompressible Navier-Stokes equations when a global attractor was shown to exist with a sharp estimate for its dimension (Constantin, Foias and Temam (1988)), with further estimates on the number of determining modes and nodes : see Foias and Prodi (1967), Foias and Temam (1984), Jones and Titi (1993), Olson and Titi (2003), Farhat, Jolly and Titi (2014).…”

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations

“…However, a proper and rigorous framework for the nudging approach was recently developed in [6], where the authors consider a more general setting which is valid for a broad class of infinite-dimensional dissipative PDEs and observables. Although the results in [6] are obtained for the two-dimensional incompressible Navier-Stokes equations as the reference model and under the assumption of continuous in time and error-free measurements, later works applied this method to several other dissipative dynamical systems [2,10,24,25,26,33,59], as well as to more general situations such as discrete in time and error-contaminated measurements ( [8,35]). Moreover, this method has also been shown to perform remarkably well in numerical simulations [3,23,44,48].…”