Input-output analysis of stochastic base flow uncertainty

We adopt an input-output approach to analyze the effect of persistent white-in-time structured stochastic base flow perturbations on the mean-square properties of the linearized Navier-Stokes equations. Such base flow variations enter the linearized dynamics as multiplicative sources of uncertainty that can alter the stability of the linearized dynamics and their receptivity to exogenous excitations. Our approach does not rely on costly stochastic simulations or adjoint-based sensitivity analysis. We provide verifiable conditions for mean-square stability and study the frequency response of the flow subject to additive and multiplicative sources of uncertainty using the solution to the generalized Lyapunov equation. For small-amplitude base flow perturbations, we bypass the need to solve large generalized Lyapunov equations by adopting a perturbation analysis. We use our framework to study the destabilizing effects of stochastic base flow variations in transitional parallel flows, and the reliability of numerically estimated mean velocity profiles in turbulent channel flows. We uncover the Reynolds number scaling of critically destabilizing perturbation variances and demonstrate how the wall-normal shape of base flow modulations can influence the amplification of various length scales. Furthermore, we explain the robust amplification of streamwise streaks in the presence of streamwise base flow variations by analyzing the dynamical structure of the governing equations as well as the Reynolds number dependence of the energy spectrum.

Figure 1

Side view of the three-dimensional canonical flows considered in this study along with various realizations of stochastic base flow perturbations ${\gamma }_u\left(t\right)$ represented by the shaded area surrounding the base flow profiles. (a) Couette flow; (b) Poiseuille flow; and (c) turbulent channel flow.

Figure 2

(a) A shape function $f\left(y\right)$ determined using Eq. (6) with $\{y_1,y_2\}=\{-0.9,0.9\}$ and $a=200$; and (b) a two-sided shape function $f\left(y\right)=f_1\left(y\right)+f_2\left(y\right)$ in which $f_1\left(y\right)$ and $f_2\left(y\right)$ are determined using Eq. (6) with $\{y_1,y_2\}=\{-1,-0.95\}$ for $f_1\left(y\right)$, $\{y_1,y_2\}=\{0.95,1\}$ for $f_2\left(y\right)$, and $a=200$.

Figure 3

Linear fractional transformation of an LTI system subject to both additive and multiplicative stochastic disturbances [Eqs. (12)]. Here, $d\tilde{\mathbf{f}}$, $d{\tilde{\gamma }}_u$, and $d{\tilde{\gamma }}_w$ represent differentials of Wiener processes that model additive and multiplicative sources of stochastic uncertainty.

Figure 4

Stability curves for fluctuation dynamics with $\mathbf{k}=(1,1)$ in (a) Couette flow; and (b) Poiseuille flow. The curves demonstrate the Reynolds number dependence of the maximum tolerable variance for stochastic base flow perturbations entering the dynamics through the shape functions $f\left(y\right)$ corresponding to Eq. (7) ($+$) or Figs. 2 ($*$) and 2 ($\circ$). For a given Reynolds number, the shaded areas under the curves denote the variances of stochastic base flow uncertainty that do not violate MSS ($\rho \left(\mathbb{L}\right)<1$ with $\alpha =1$). The triangles in the upper right corners demonstrate an $R^{-1}$ slope.

Figure 5

Logarithmically scaled critical variance levels for stochastic multiplicative uncertainty ${\overline{\gamma }}_u$ with $\alpha =1$ and $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$ over the horizontal wavenumber spectrum in (a) Couette flow with $R=500$ and (b) Poiseuille flow with $R=2000$.

Figure 6

Energy spectra of (a) plane Couette flow with $R=500$ and (b) plane Poiseuille flow with $R=2000$. Color plots show ${log}_{10}\left[E_0\left(\mathbf{k}\right)\right]$.

Figure 7

The discounted energy spectrum $E_c$ in Couette flow with $R=500$ and $k_x=1$ subject to base flow perturbations with $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$ with ${\sigma }_u^2=0.43$ and $\alpha =0.01$. Direct solution from solving Eq. (20) ($-$); and approximate solutions from perturbation analysis: $E_c={\alpha }^2{E}_2\left(\mathbf{k}\right)$ ($*$) and $E_c={\alpha }^2{E}_2\left(\mathbf{k}\right)+{\alpha }^4{E}_4\left(\mathbf{k}\right)$ ($\circ$).

Figure 8

The second-order correction to the energy spectrum $E_2\left(\mathbf{k}\right)$ due to multiplicative uncertainty ${\overline{\gamma }}_u$ in Couette flow with $R=500$ (left); and Poiseuille flow with $R=2000$ (right). Shape functions $f\left(y\right)$ correspond to (a), (b) Eq. (7); (c), (d) Fig. 2; (e), (f) Fig. 2. Variances ${\sigma }_u^2$: (a) $4.38\times {10}^{-3}$; (b) $6.25\times {10}^{-4}$; (c) 0.004; (d) $6.25\times {10}^{-4}$; (e) 0.005; (f) 0.001. Color plots show ${log}_{10}\left[E_2\left(\mathbf{k}\right)\right]$. The symbols ($\times$) and ($•$) mark the wavenumber pairs associated with oblique waves and TS waves, respectively.

Figure 9

The correction to the energy spectrum $E_c\left(\mathbf{k}\right)$ due to multiplicative uncertainty ${\overline{\gamma }}_u$ with $\alpha =0.5$ in Couette flow with $R=500$ (left); and Poiseuille flow with $R=2000$ (right). Shape functions $f\left(y\right)$ correspond to (a), (b) Eq. (7); (c), (d) Fig. 2; (e), (f) Fig. 2. Variances ${\sigma }_u^2$: (a) $4.38\times {10}^{-3}$; (b) $6.25\times {10}^{-4}$; (c) 0.004; (d) $6.25\times {10}^{-4}$; (e) 0.005; (f) 0.001. Color plots show ${log}_{10}\left[E_c\left(\mathbf{k}\right)\right]$.

Figure 10

The relative correction to the turbulent kinetic energy ${\int }_{\mathbf{k}}{E}_c\left(\mathbf{k}\right)d\mathbf{k}/{\int }_{\mathbf{k}}{E}_0\left(\mathbf{k}\right)d\mathbf{k}$ in (a) Couette flow with $R=500$; and (b) Poiseuille flow with $R=2000$ subject to base flow variations with amplitude $\alpha$. The curves demonstrate the $\alpha$ dependence of the energy correction due to stochastic base flow perturbations entering the dynamics through $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$ ($+$), Fig. 2 ($*$), and Fig. 2 ($\circ$). (a) Base flow perturbations are introduced with variances of ${\sigma }_u^2=4.38\times {10}^{-3}$ ($+$), 0.004 ($*$); and 0.005 ($\circ$) into Couette flow, and (b) ${\sigma }_u^2=6.25\times {10}^{-4}$ ($+$), $6.25\times {10}^{-4}$ ($*$); and 0.001 ($\circ$) into Poiseuille flow.

Figure 11

Contribution of the first six eigenvalues of the velocity covariance matrix $\mathrm{\Phi }$ of channel flow in the absence ($*$) and presence ($\circ$) of base flow perturbations with $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$ and amplitude $\alpha =1$. (a) Couette flow with $R=500$ at $\mathbf{k}=(0.95,2.29)$; and (b) Poiseuille flow with $R=2000$ at $\mathbf{k}=(0.38,3.02)$. The variance of base flow uncertainties: (a) ${\sigma }_u^2=0.50$; (b) ${\sigma }_u^2=0.21$.

Figure 12

The streamwise component of the dominant flow structures of Couette flow with $R=500$ and $\mathbf{k}=(0.95,2.29)$ in the absence (first rows) and presence (second rows) of stochastic base flow perturbations with $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$, $\alpha =1$, and ${\sigma }_u^2=0.50$; (a) the principal modes; and (b) the second most energetic modes. The three columns correspond to: (left) the spatial structure of the eigenmodes with red and blue colors denoting regions of high and low velocity; (middle) the streamwise velocity at $z=0$; and (right) the $y\text{−}z$ slice of streamwise velocity (color plots) and vorticity (contour lines) at the streamwise location indicated by the dashed vertical lines in the middle panel.

Figure 13

The streamwise component of the dominant flow structures of Poiseuille flow $R=2000$ and $\mathbf{k}=(0.38,3.02)$ in the absence (first rows) and presence (second rows) of stochastic base flow perturbations with $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$, $\alpha =1$, and ${\sigma }_u^2=0.21$; (a) the principal modes; and (b) the second most energetic modes. The three columns correspond to: (left) the spatial structure of the eigenmodes with red and blue colors denoting regions of high and low velocity; (middle) the streamwise velocity at $z=0$; and (right) the $y\text{−}z$ slice of streamwise velocity (color plots) and vorticity (contour lines) at the streamwise location indicated by the dashed vertical lines in the middle panel.

Figure 14

(a) Stability curves for fluctuation dynamics with $\mathbf{k}=(2.5,7)$ in turbulent channel flow subject to stochastic base flow perturbations following shape functions $f\left(y\right)$ corresponding to Eq. (7) ($+$) and those shown in Figs. 2 ($*$) and 2 ($\circ$). The shaded areas under the curves denote the variances of stochastic base flow uncertainty that do not violate MSS $\left[\rho \right(\mathbb{L})<1$ with $\alpha =1]$. The triangle in the upper right corner demonstrates an $R_{\tau }^{-1}$ slope. (b) Logarithmically scaled critical variance levels of stochastic multiplicative uncertainty ${\overline{\gamma }}_u$ with $\alpha =1$ and $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$ over the horizontal wavenumber spectrum of turbulent channel flow with $R_{\tau }=186$.

Figure 15

(a) Premultiplied energy spectrum of turbulent channel flow with $R_{\tau }=186$ in the absence of stochastic base flow perturbations $\left[E_0\left(\mathbf{k}\right)\right]$. (b)–(g) Corrections to the premultiplied energy spectra $E_c\left(\mathbf{k}\right)$ of turbulent channel flow $R_{\tau }=186$ due to stochastic multiplicative uncertainty ${\gamma }_u$ with variance ${\sigma }_u^2=0.13$ and perturbation amplitudes $\alpha =0.05$ (second row) and $\alpha =0.9$ (third row) that follow perturbations shapes $f\left(y\right)$ corresponding to Eq. (7) (b), (e), Fig. 2 (c), (f), and Fig. 2 (d), (g).

Figure 16

The total effect of stochastic perturbations of amplitude $\alpha$ on the energy spectrum of turbulent channel flow with $R_{\tau }=186$. The curves demonstrate the $\alpha$ dependence of the energy correction due to base flow perturbations entering the dynamics through the shape functions $f\left(y\right)$ corresponding to Eq. (7) ($+$), in addition to those depicted in Figs. 2 ($*$) and 2 ($\circ$).

Figure 17

Contribution of the first eight eigenvalues of the velocity covariance matrix $\mathrm{\Phi }$ of channel flow in the absence ($*$), and presence ($\circ$) of base flow perturbations with $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$ and amplitude $\alpha =1$ in turbulent channel flow with $R_{\tau }=186$ at $(k_x,k_z)=(1.86,1.94)$.

Figure 18

The streamwise component of the dominant flow structures of turbulent channel flow with $R_{\tau }=186$ and $(k_x,k_z)=(1.86,1.94)$ in the absence (a) and presence (b) of stochastic base flow perturbations of amplitude $\alpha =1$, shape $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$, and variance ${\sigma }_u^2=0.49$. The three columns correspond to: (left) the spatial structure of the eigenmodes with red and blue colors denoting regions of high and low velocity; (middle) the streamwise velocity at $z=0$; and (right) the $y\text{−}z$ slice of streamwise velocity (color plots) and vorticity (contour lines) at the streamwise location indicated by the dashed vertical lines in the middle panel.

Figure 19

The $k_z$-dependence of functions (a) $f$, (b) $h$, and (c) $g$ in Eq. (29) for Couette flow with $R=500$ (—) and Poiseuille flow with $R=2000$ ($—$) subject to base flow perturbations of shape $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$ of variance ${\sigma }_u^2=1.13\times {10}^5$ and ${\sigma }_u^2=3.22\times {10}^3$, respectively. The function $f$, which is responsible for the $O\left(R\right)$ energy amplification is the same for both Couette and Poiseuille flows.

Figure 20

Logarithmically scaled terms that are responsible for the $O\left(R^2\right)$ energy amplification in Eq. (29) (${log}_{10}\left[g(k_z,{\sigma }_u^2)\right]$) as a function of spanwise wavenumber $k_z$ and base flow perturbation variance ${\sigma }_u^2$. Perturbations to the base flow follow $f\left(y\right)=\overline{U}\left(y\right)/max\left(\right|\overline{U}\left(y\right)\left|\right)$. (a) Couette flow with $R=500$; and (b) Poiseuille flow with $R=2000$.