Optimally amplified large-scale streaks and drag reduction in turbulent pipe flow

The optimal amplifications of small coherent perturbations within turbulent pipe flow are computed for Reynolds numbers up to one million. Three standard frameworks are considered: the optimal growth of an initial condition, the response to harmonic forcing and the Karhunen-Loève (proper orthogonal decomposition) analysis of the response to stochastic forcing. Similar to analyses of the turbulent plane channel flow and boundary layer, it is found that streaks elongated in the streamwise direction can be greatly amplified from quasistreamwise vortices, despite linear stability of the mean flow profile. The most responsive perturbations are streamwise uniform and, for sufficiently large Reynolds number, the most responsive azimuthal mode is of wave number $m=1$. The response of this mode increases with the Reynolds number. A secondary peak, where $m$ corresponds to azimuthal wavelengths ${\lambda }_{\theta }^{+}\approx 70–90$ in wall units, also exists in the amplification of initial conditions and in premultiplied response curves for the forced problems. Direct numerical simulations at $\text{Re}=5300$ confirm that the forcing of $m=1,2$ and $m=4$ optimal structures results in the large response of coherent large-scale streaks. For moderate amplitudes of the forcing, low-speed streaks become narrower and more energetic, whereas high-speed streaks become more spread. It is further shown that drag reduction can be achieved by forcing steady large-scale structures, as anticipated from earlier investigations. Here the energy balance is calculated. At $\text{Re}=5300$ it is shown that, due to the small power required by the forcing of optimal structures, a net power saving of the order of 10% can be achieved following this approach, which could be relevant for practical applications.

Figure 1

(Color online) Optimal energy growth of initial conditions $G_{\text{max}}$ (a), optimal response to harmonic forcing $R_{\text{max}}$ (b) and variance of the response to stochastic forcing $V$ (c) for ${\text{Re}}_{\tau }=19200$. Dependence on azimuthal wave number $m$, with corresponding wavelength in wall units ${\lambda }_{\theta }^{+}$ (top axis), for selected values of the dimensionless streamwise wave number $\alpha$. Reference slopes $m^{-2}$ and $m^{-1}$ (dotted lines) are shown in subfigures (b) and (c) respectively. The dependence of the optimal harmonic forcing amplification $R\left(\alpha ,m=1;{\Omega }_f\right)$ on the forcing frequency ${\Omega }_f$ is reported in the insert of subfigure (b) for $\alpha =0$ and $\alpha =1$; $R_{\text{max}}={\text{max}}_{{\Omega }_f}R$. The percentage contribution to the variance, $100{\sigma }_j/V\left(0,1\right)$, of the response to stochastic forcing contained in the first 20 Karhunen-Loève modes is reported in the insert of subfigure (c).

Figure 2

(Color online) Dependence on the azimuthal wave number $m$ and the corresponding azimuthal wavelength (top axis) of: the optimal energy amplification of initial condition $G_{\text{max}}$ [(a) and (d)], the premultiplied optimal response to harmonic forcing $m^2{R}_{\text{max}}$ [(b) and (e)] and of the premultiplied variance of the response stochastic forcing $mV$ [(b) and (e)]. Streamwise uniform $\left(\alpha =0\right)$ perturbations are considered for the selected Reynolds numbers $\text{Re}={10}^4,{10}^5$ and ${10}^6$ corresponding to ${\text{Re}}_{\tau }=317$, 2830, and 19200. The responses, wave numbers and wavelengths are scaled in outer units on the upper row [(a), (b), and (c)] and in inner units in the lower row [(d), (e), and (f)].

Figure 3

(Color online) Linear optimal responses to steady forcing for $\alpha =0$ and $m=1,2,4$; ${\text{Re}}_{\tau }=19200$. Vectors: cross-stream components of the input vector field. (Colored) contour levels: streamwise component of the output field (streaks). White fast-streaks, red/dark slow streaks.

Figure 4

(Color online) Response to forcing in full simulation showing the linear régime and nonlinear saturation. $\Vert \mathit{f}\Vert$ is the rms value. The isolated line is of slope 1 and squares correspond to the plots of Fig. 5.

Figure 5

(Color online) Nonlinear responses to finite amplitude forcing for $\alpha =0$ and $m=1,2,4$ from direct numerical simulation at ${\text{Re}}_{\tau }=180$. Small forcing amplitude (upper row) and moderate forcing (lower row), corresponding to the marked squares in Fig. 5. (a) $m=1$, $\Vert f\Vert ={8.010}^{-5}$; (b) $m=2$, $\Vert f\Vert ={1.410}^{-5}$; (c) $m=4$, $\Vert f\Vert ={8.010}^{-5}$; (d) $m=1$, $\Vert f\Vert ={8.010}^{-4}$; (e) $m=2$, $\Vert f\Vert ={8.010}^{-4}$; (f) $m=4$, $\Vert f\Vert ={2.510}^{-4}$. Vectors: cross-stream components of the forced velocity field. (Colored) contour levels: streamwise component of the output field (streaks). White fast-streaks, red/dark slow-streaks.

Figure 6

(Color online) Relative drag for several levels of steady forcing of the $m=1$ mode from direct numerical simulation. At intermediate levels of forcing the drag is reduced, falling below the average of the unforced case (horizontal straight line). The forcing can reduce the drag. The solid line is a higher resolution check at $\Vert f\Vert ={8.010}^{-4}$.

Figure 7

(Color online) Effect of forcing on the mean skin friction, relative to the unforced case (solid). Relative net power consumption including the work done by the force (dashed). Black square corresponds to $\Vert f\Vert ={8.010}^{-4}$.

Figure 8

(Color online) Snapshots of the cross-stream distribution of the streamwise vorticity ${\omega }_x$ field at a selected streamwise station from direct numerical simulation. (a) Streamwise vorticity in the absence of forcing. (b,c) Weakened and localized streamwise vorticity in the presence of the $m=1,2$ steady forcing, corresponding to optimal drag reduction. Blue/dark intense vorticity, white no vorticity. Same color scales are used on all the subplots. Contour lines denote isolevels of the streamwise averaged velocity (the forced streaks).