From limited observations to the state of turbulence: Fundamental difficulties of flow reconstruction

Numerical simulations of turbulence provide nonintrusive access to all the resolved scales and any quantity of interest, although they often invoke idealizations and assumptions that can compromise realism. In contrast, experimental measurements probe the true flow with lesser idealizations, but they continually contend with spatiotemporal sensor resolution. Assimilating observations directly in simulations can combine the benefits of both approaches and mitigate their respective deficiencies. The problem is expressed in variational form, where we seek the flow field that satisfies the Navier-Stokes equations and minimizes a cost function defined in terms of the deviation of the computational predictions and available observations. In this framework, measurements are no longer a mere record of the instantaneous, local quantity, but rather an encoding of the antecedent flow events that we aim to decode using the governing equations. Chaos plays a central role in obfuscating the interpretation of the data: observations that are infinitesimally close may be due to entirely different earlier conditions. We examine a number of state estimation problems: In circular Couette flow, starting from observations of the wall stress, we accurately reconstruct the wavy Taylor vortices that interact nonlinear to maintain a saturated state. Through a discussion of transition to chaos in a Lorenz system, we highlight the challenge of navigating the landscape of the cost function, and how the landscape can favorably be modified by sensor weighting and placement. In turbulent channel flow, the Taylor microscale and Lyapunov timescale place restrictions on the resolution of observations for which we can accurately reconstruct all the missing scales. The notion of domain of dependence of an observation is introduced and related to the Hessian of the cost function. For measurements of wall shear stress, the eigenspectrum of the Hessian demonstrates the sensitivity of short-time observations to the fine near-wall turbulent scales and to the large scales only in the outer flow. At long times, backward chaos obfuscates the interpretation of the data: observations that are infinitesimally close may be traced back to entirely different earlier flow states—a dual to the more common butterfly effect for forward trajectories.

This article appears in the following collection:

Figure 1

Schematic of adjoint-variational data assimilation. Starting from an estimate of the initial condition ${\mathit{u}}_0$, we compute the Navier-Stokes evolution. The forward fields ${\mathit{u}}_n$ and the deviation from observations ${\mathcal{E}}_n$ are stored. The adjoint equations are evolved from the final to the initial time, driven by observations errors. The initial adjoint field ${\mathit{u}}_0^{†}$ is the gradient of the cost function, and is used to update ${\mathit{u}}_0$. The forward-adjoint loop is repeated until convergence.

Figure 2

Energy spectra averaged across the gap, ${log}_{10}\langle \parallel \widehat{\mathit{u}}{{\parallel }^2\rangle }_d$, where $\widehat{\mathit{u}}$ is the Fourier transform of the velocity perturbation field $\mathit{u}-\langle {\mathit{u}}^R\rangle$. Insets: isosurfaces of (red) positive and (blue) negative azimuthal velocity fluctuations, $v_{\theta }-\langle v_{\theta }^R\rangle =\{0.14,-0.14\}$. (a) Adjoint-variational estimation at $t=T$; (b) the true state. Both spectra are normalized by the maximum value of the true spectrum.

Figure 3

(a) Evolution of the Lorenz system in physical space. Contours are the temperature perturbations and arrows show the velocity field. (b) State-space evolution of the Fourier coefficients, starting from the true initial condition with marked observations. (c) Time evolution of $Y$ from the (gray) true and (blue) estimated initial conditions, with marked observations. The assimilated trajectory latches onto the truth, reproduces observations, and accurately forecasts the system for some time beyond the observation horizon [18].

Figure 4

The dependence of the cost function of the Lorenz system on $(X_o,Y_o)$ near the global optimal. Left: The landscape of the cost function when all observations are uniformly weighted. Right: The landscape of $J$ when the observations are weighted by a Gaussian that preferentially weights early measurements.

Figure 5

State-space evolution from estimates of the initial Fourier coefficients of the Lorenz system. (a) Divergence of trajectories starting from the true and a slightly perturbed initial condition. (b) Trajectory from estimated initial condition. Despite a large initial separation from the truth, the assimilated trajectory latches onto the truth, reproduces observations, and accurately forecasts the system for some time beyond the observations horizon.

Figure 6

(a) True vortical structures visualized using the ${\lambda }_2$ vortex identification criterion, with threshold ${\lambda }_2=-2$. The isosurfaces are colored by wall-normal height, $y$. Pixels on the black cross-flow planes represent scarce observations, or subsampled velocity data. (b) Vortical structures computed from interpolation of the observations.

Figure 7

(a) Temporal evolution of volume-averaged error of the adjoint-variational estimated velocity field. Black line: initial condition at $t=0$ is estimated using data within $t\in [0,T]$; Green line: flow state at $t=2T$ is estimated using data within $t\in [2T,3T]$. (b) Horizontally averaged error ${\mathcal{E}}_{xz}\left({\omega }_z^{+}\right)$ in spanwise vorticity field at $t=T$. (Solid) Adjoint-variational estimation was performed using velocity data that is (black) noise-free and (blue) contaminated by 10% noise relative to the local velocity. (Blue dashed) Vorticity is evaluated directly from interpolation of noisy velocity data. Errors are normalized by mean vorticity $\langle \partial u/\partial y\rangle$ at the wall. (c, d) Estimated and true vortical structures at $t=T$, visualized using the ${\lambda }_2$ vortex identification criterion, with threshold ${\lambda }_2=-2$ and colored by wall-normal distance $y$.

Figure 8

Schematic of forward-adjoint duality (15) in turbulent channel flow. Top: Forward evolution of an initial perturbation, with isosurfaces showing the streamwise velocity $u^{\prime }$, colored by wall-normal distance $y$. The associated impact on the observation is evaluated at the wall at $t^{+}=t_m^{+}=20$. Bottom: Backward evolution of the adjoint field starting from an impulse at the wall, up to $\tau =t_m$, where isosurfaces mark (red) positive and (blue) negative adjoint streamwise velocities $u^{†}$. Modification to the observation due to any initial forward perturbation can be efficiently computed from its inner product with this single adjoint field at $t=0$ ($\tau =t_m$).

Figure 9

(a) The largest eigenvalue of the Hessian matrix $\widehat{\mathcal{H}}(k_x,k_z;t_m)$ for observing ${\partial u/\partial y|}_{\mathrm{wall}}$ instantaneously at $t_m^{+}=20$. Color and line contours are for ${\text{Re}}_{\tau }=180$ and 590, respectively. The eigenvalues are normalized by the supremum. (b) Visualizations of the leading eigenfunctions of the Hessian at selected wave-number pairs: $(k_x^{+},k_z^{+})$ = (0.08, 0), (0, 0.14), (0, 0.02) for modes $I_A$, $I_B$ and $II$. Contours show the velocity magnitude and arrows are in-plane velocity components.