Identifying the body force from partial observations of a two-dimensional incompressible velocity field

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.

Figure 1

(a) Visualization of reference force and initial data, (bottom-left cut-away) $-{\nabla }^2{f}^{\psi }/{\parallel {\nabla }^2{f}^{\psi }\parallel }_{L^{\infty }}$, (top-right cut-away) $-{\nabla }^2{\psi }_0/{\parallel {\nabla }^2{\psi }_0\parallel }_{L^{\infty }}$. (b) Weighted spectra of force $f^{\psi }$ and initial data ${\psi }_0$. Initial data and reference force. In (a), the normalized initial vorticity $-{\nabla }^2{\psi }_0/{\parallel {\nabla }^2{\psi }_0\parallel }_{L^{\infty }}$ is shown rather than the initial stream function ${\psi }_0$, to highlight details. Similarly, $-{\nabla }^2{f}^{\psi }/{\parallel {\nabla }^2{f}^{\psi }\parallel }_{L^{\infty }}$ is shown in (a) rather than $f^{\psi }$. For reference, we computed that $\parallel {\nabla }^2{\psi }_0{\parallel }_{L^{\infty }}\approx 2.8256$ and $\parallel {\nabla }^2{f}^{\psi }{\parallel }_{L^{\infty }}\approx 0.7443$. In (b), the weighted spectra of the initial data and stream function are shown.

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 a lack of convergence.

Figure 3

Log-linear plot of $L^2$ 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 of the spectra of (a) the error in the stream function and (b) the error of the forcing function at times $t=0.0,0.5,1.0$, and also $t=2,4,6...,40$ using the direct replacement scheme. Colors move from blue ($t=0.0$) to red ($t=40.0)$. The vertical dashed line is the observational wave-number cutoff at ${\left|\mathbf{k}\right|}_{\infty }=64$. The horizontal dotted line is machine precision ($\epsilon \approx 2.22\times {10}^{-16}$). (c) Relative error in spectrum spectrum at times $t=0.0,0.5,1.0$, and also $t=2,4,6...,80$ using the direct replacement scheme. (d) Relative error in forcing spectrum at times $t=0.0,0.5,1.0$, and also $t=2,4,6...,80$ using the direct replacement scheme.