Uncertainty Quantification in Resolvent Analysis of Experimental Wall-Bounded Turbulent Flows

TL;DR

This paper tackles the problem of quantifying how uncertainties in near-wall mean-flow measurements affect resolvent analysis of wall-bounded turbulence. It develops a first-order sensitivity framework that computes perturbations in resolvent gains $\sigma_i$ and corresponding modes $\boldsymbol{\psi}_i$, $\boldsymbol{\phi}_i$ using perturbations in the mean flow $\Delta\overline{\boldsymbol{U}}$ with essentially zero additional computational cost. The authors derive explicit expressions, e.g., $\partial\sigma_i = -\sigma_i^2 \mathrm{Re}(\langle \boldsymbol{\phi}_i, \mathbf{W}_f \partial \mathbf{L} \boldsymbol{\psi}_i \rangle)$, and validate them in both local (1D) and biglobal (2D) settings, including PIV-based mean fields with different near-wall fits. They show that near-wall inaccuracies can produce significant changes in $\sigma_1$ and alter the inferred flow physics, particularly for near-wall small-scale modes and large-scale biglobal structures, emphasizing the need for high-resolution near-wall data or robust error bounding. The study offers practical experimental design guidance and demonstrates how the sensitivity framework can bound resolvent-errors when near-wall measurements are limited or uncertain, thereby enhancing the reliability of resolvent-based predictive tools for wall-bounded flows.

Abstract

Experimental mean flows are commonly used to study wall-bounded turbulence. However, these measurements are often unable to resolve the near-wall region and thus introduce ambiguity in the velocity closest to the wall. This poses a source of uncertainty in equation-based approaches that rely on these mean flow measurements such as resolvent analysis. Resolvent analysis provides a scale-dependent decomposition of the linearized Navier Stokes equations that identifies optimal gains, response modes and forcing modes that has been used to great effect in turbulent wall-bounded flows. Its potential in the development of predictive tools for a variety of wall-bounded flows is high but the limitations of the input data must be addressed. Here, we quantify the sensitivity of resolvent analysis to common sources of experimental uncertainty and show that this sensitivity can be quantified with minimal additional computational cost. This approach is applied to both local and biglobal resolvent analysis by using artificial disturbances to mean profiles in the former and particle image velocimetry measurements with differing near-wall fits in the latter. We also highlight an example where poor near-wall resolution can lead to erroneous conclusions compared to the full-resolution data in an adverse pressure gradient turbulent boundary layer.

Figures (6)

• Figure 1: (a) $\overline{U}^{+}$ vs $y^{+}$ for the unperturbed flow $\overline{U}_{0}$ (black), perturbation to $u_{\tau}$ (red), and offset in $y$ (blue). (b) difference in perturbed flow fields and unperturbed flows with thin black lines representing $\pdv*{\overline{U}}{a}\Delta a$. (c) percent difference in perturbed flow fields and unperturbed flows.
• Figure 2: Comparison of 1D $\hbox{Re}(\psi_{u,1})$ (a) $\hbox{Re}(\phi_{v,1})$ (b), and the percent difference of $\sigma_{j}$ computed from the perturbed flows relative to the unperturbed $\overline{U}$ (c) computed from unperturbed and perturbed $\overline{U}$. The colors are the same as in Figure \ref{['fig_MeanFlowFields']}. The predicted perturbed $\sigma_{j}$, $\sigma_{j} + \Delta\sigma_{j}$, are plotted in black circles in (c). Comparison of $\left\langle\boldsymbol{\psi}_{j},\mathbf{W}_{r}\Delta\boldsymbol{\psi}_{i} \right\rangle$ using the prediction (d) and actual difference (e) for a perturbation in $\Delta y$. Comparison of $\Delta \lambda_{j}$ (red squares) and $\Delta \sigma_{1}/\sigma_{1}^{2}$ (red circles) against $\Delta u_{\tau}$ where $j$ is chosen as the $\hbox{Re}(\lambda^{+}_{j}/k_{x}^{+})$ closest to $10$ in (f). The solid black lines are the predicted $\Delta \lambda_{j}$ and $\Delta \sigma_{1}/\sigma_{1}^{2}$. The resolvent analysis and stability analysis are computed with $k_{x}^{+} = 2\pi/1000$ and $k_{z}^{+} = 2\pi/100$ and the former uses $\omega^{+} = 10 k_{x}^{+}$.
• Figure 3: Contours of $\sigma_{1}^{+}k_{x}^{+}k_{z}^{+}$ and $k_{x}^{+}k_{z}^{+}\partial\sigma_{1}^{+}/\partial u_{\tau}$ for fixed $\omega^{+}/k_{x}^{+} = 10$ ($y_{c}^{+} =13$) (a,d), $\omega^{+}/k_{x}^{+} = 18$ ($y_{c}^{+} = 200$) (b,e), and $\omega^{+}/k_{x}^{+} = 23$ ($y_{c}^{+} = 750$) (c,f). The colored contours plot $k_{x}^{+}k_{z}^{+}\abs{\partial\sigma_{1}^{+}/\partial u_{\tau}}$ while the solid contour lines are positive and dashed contours are negative in (d--f). The black contours are computed using the derivatives and the blue are computed using finite differences. The contour lines are in increments of $.1$ and $.02$ in (d) and (e,f), respectively.
• Figure 4: Contour of $\hbox{Re}(\psi_{u,1})$ (a). The percent difference in $\overline{U}$ by fitting to a DNS profile and the van Driest near-wall fit (b). Comparison of $\left\langle\boldsymbol{\psi}_{j},\mathbf{W}_{r}\Delta\boldsymbol{\psi}_{i} \right\rangle$ using the prediction (c) and actual difference (d) when using the two different near-wall fits. The percent difference in $\Delta \sigma_{j}$ computed from the prediction (black circles) and the true difference (red circles).
• Figure 5: Contours of $k_{z}^{+}\omega^{+}\sigma_{1}^{+}$ (a), $k_{z}^{+}\omega^{+}\Delta\sigma_{1}^{+}$ (b) and $\Delta\sigma_{1}/\sigma_{1} (\%)$ (c) for the PIV data with different near-wall fits. The solid and dashed contours correspond to positive and negative values spaced at increments of $.065$ in (b) and $1.5$ in (c). The blue contours are computed using finite differences.