Reconstructing the time evolution of wall-bounded turbulent flows from non-time-resolved PIV measurements

Particle image velocimetry (PIV) systems are often limited in their ability to fully resolve the spatiotemporal fluctuations inherent in turbulent flows due to hardware constraints. In this study, we develop models based on rapid distortion theory (RDT) and Taylor's hypothesis (TH) to reconstruct the time evolution of a turbulent flow field in the intermediate period between consecutive PIV snapshots obtained using a non-time resolved system. The linear governing equations are evolved forward and backward in time using the PIV snapshots as initial conditions. The flow field in the intervening period is then reconstructed by taking a weighted sum of the forward and backward estimates. This spatiotemporal weighting function is designed to account for the advective nature of the RDT and TH equations. Reconstruction accuracy is evaluated as a function of spatial resolution and reconstruction time horizon using direct numerical simulation data for turbulent channel flow from the Johns Hopkins Turbulence Database. This method reconstructs single-point turbulence statistics well and resolves velocity spectra at frequencies higher than the temporal Nyquist limit of the acquisition system. Reconstructions obtained using a characteristics-based evolution of the flow field under TH prove to be more accurate compared to reconstructions obtained from numerical integration of the discretized forms of RDT and TH. The effect of measurement noise on reconstruction error is also evaluated.

Figure 1

Left: Non-time-resolved PIV snapshots stacked along the time axis. Right: Time-resolved reconstruction of the velocity field from the PIV snapshots.

Figure 2

Schematic showing the combined Region of Influence (ROI) and Domain of Dependence (DOD) of two consecutive snapshots in the $x\text{−}t$ plane at a given wall-normal location. Characteristics with slope determined by the local mean velocity $[dt/dx=1/U(y\left)\right]$ are shown by the solid black lines at the yellow-green and green-blue interfaces. The green region shows where the ROI of Snapshot 1 (S1) and the DOD of Snapshot 2 (S2) coincide. In this region both the forward and backward estimates are used for fusion.

Figure 3

Time variation of integrated reconstruction error for the benchmark case with $T^{+}\approx 16$ and $\mathrm{\Delta }{x}^{+}=\mathrm{\Delta }{y}^{+}\approx 16$. The error is computed using Eq. (12) and averaged over 48 different realizations. The solid, dashed, and dotted lines represent the mean and the color bands are spaced at 1 standard deviation from the mean line. Table 2 provides a description of the different reconstruction techniques used in this figure.

Figure 4

Wall-normal variation in reconstruction error as a function of time for the (a) ${\text{RDT}}_{tx}^{\pm }$ (b) ${\text{TH}}_{tx}^{\pm }$ and (c) ${}^{*}{\text{TH}}_{tx}^{\pm }$ models. The error is calculated using Eq. (13) and averaged over 48 different realizations.

Figure 5

Snapshots of the velocity field from DNS (a), (b), the ${\text{RDT}}_{tx}^{\pm }$ reconstruction (c), (d), and the ${}^{*}{\text{TH}}_{tx}^{\pm }$ reconstruction (e), (f) in the middle of the time horizon, when error is maximum. Profiles of streamwise and wall-normal velocity at $y^{+}\approx 200$ are plotted in panels (g) and (h). The bold solid lines show the reconstructed velocity field for ${\text{RDT}}_{tx}^{\pm }$, dotted lines are for ${}^{*}{\text{TH}}_{tx}^{\pm }$, and the fine solid lines show the DNS velocity field. The locations of the shocks in the weighting functions for the reconstruction are highlighted in red.

Figure 6

Maximum reconstruction error (${\epsilon }_{\text{max}}$) as a function of the grid resolution ($\mathrm{\Delta }{x}^{+}$) and time horizon $T^{+}$ for (a) ${\text{RDT}}_{tx}^{\pm }$, (b) ${\text{TH}}_{tx}^{\pm }$, and (c) ${}^{*}{\text{TH}}_{tx}^{\pm }$. The contour lines are shown at intervals of 0.05, and the benchmark case is shown as the black cross. Figure 7 shows the variation in integrated error as a function of time for the case marked with a black square ($\mathrm{\Delta }{x}^{+}=4;T^{+}=32$).

Figure 7

Variation in integrated error Eq. (12) as a function of time for the RDT, TH, and ${}^{*}\mathrm{TH}$ reconstructions at the grid resolution and time horizon marked with a black square ($\mathrm{\Delta }{x}^{+}=4;T^{+}=32$) in Fig. 6.

Figure 8

Comparison of wall-normal profiles of (a) $\overline{u^2}$, (b) $\overline{v^2}$, and (c) $\overline{uv}$ from DNS with the ${\text{TH}}_{tx}^{\pm }$ and ${}^{*}{\text{TH}}_{tx}^{\pm }$ reconstructions. The solid black lines show statistics computed directly from DNS data. The dashed lines show statistics for ${\text{TH}}_{tx}^{\pm }$ and the dotted lines show statistics for ${}^{*}{\text{TH}}_{tx}^{\pm }$. Reconstructions were carried out with the DNS data subsampled at intervals of $T^{+}\approx 16$.

Figure 9

Premultiplied power spectral density for streamwise velocity $f{E}_{uu}/u_{\tau }^2$ (a)–(c) and wall-normal velocities $f{E}_{vv}/u_{\tau }^2$ (d)–(f). Panels (a) and (d) are obtained from DNS data. Panels (b) and (e) show the power spectra from the PIV-like data, i.e., DNS data subsampled at a frame rate corresponding to $T^{+}\approx 16$. Panels (c) and (f) show the power spectra obtained from the characteristics-based ${}^{*}{\text{TH}}_{tx}^{\pm }$ reconstruction. Contour lines are shown at intervals of 0.2 for panels (a)–(c), and 0.04 for panels (d)–(f).

Figure 10

Error in reconstructed power spectral densities for (a) streamwise velocity and (b) wall-normal velocity. The error is computed by taking the difference between power spectra obtained from DNS and the ${}^{*}{\text{TH}}_{tx}^{\pm }$ reconstruction. The horizontal white line corresponds to the Nyquist limit for the PIV-like data sampled at $T^{+}\approx 16$. Contour lines are shown at intervals of 0.05 (a) and 0.01 (b).

Figure 11

Variation of integrated error as a function of time with 4 different initial noise levels for (a) ${\text{TH}}_{tx}^{\pm }$ and (b) ${}^{*}{\text{TH}}_{tx}^{\pm }$. The solid lines show results averaged over 48 different realizations. A shaded gray band representing one standard deviation above and below the average value from the 48 realizations is shown for the noisiest case, i.e., $\text{SNR}=5$.