Three-dimensional traveling waves in a square duct

A nonlinear streamwise traveling-wave solution is obtained by homotopy for square duct flow. For a particular symmetry of the perturbations, this wave comes into existence at about ${\text{Re}}_b=600$ (based on half-duct width and bulk speed) for a streamwise wave number $\alpha =0.85$. The resulting four-vortex mean flow resembles the transitional flow structures observed in previous simulations.

Figure 1

(Color online) The forcing function $f\left(y,z\right)$ used for forcing the mean four-vortex flow. The color coding goes from most negative (dark) to most positive (light).

Figure 2

(Color online) The continuation branch for $\text{Re}=3000$ and $\alpha =1.5$. The curve is traced out with a resolution $\left(N_x,N_y,N_z\right)=\left(4,15,15\right)$, which is acceptable. The amplitude of the TW is defined as $A_{3D}=\sqrt{{\sum }_{j=0}^{NZ}{\left|{\widehat{u}}_{10j}\right|}^2+{\left|{\widehat{v}}_{10j}\right|}^2+{\left|{\widehat{w}}_{10j}\right|}^2}$, with ${\hat{\mathit{u}}}_{10j}$ as the solution coefficients of Eq. (2). The right vertical axis shows the phase velocity $c$ (dashed) of the wave scaled by ${\overline{\widehat{U}}}_{\text{max}}$.

Figure 3

(Color online) The disturbance velocity field at the turning point (see Table ) at three stations in $x$ and the mean flow. The contour levels represent the streamwise velocity $u^{\prime }$ and range between the minimum and the maximum of $u^{\prime }$ of the chosen stations in $x$ or $-0.586$ to 0.126 in steps of 0.05. An online video provides a clear visualization of the TW [25]. The bottom right figure shows the mean flow in the shape of a four-vortex structure, similar to that observed in transitional duct flow [7]. The contour levels represent the mean streamwise velocity $\overline{u^{\prime }}$ and range between the minimum and the maximum of $\overline{u^{\prime }}$, $-0.570$ to 0.099 in steps of 0.032. Compare with Figs. 3a, 3b of Uhlmann et al. [7]. The color coding goes from most negative (dark) to most positive (light). The arrows represent the cross-stream velocity components.

Figure 4

(Color online) The skin friction $\Lambda$ (red solid curve) as a function of ${\text{Re}}_b$ for $\alpha =0.85$ and resolution (4,15,15). The plus $(+)$ represent results with resolution (6,20,20), showing good agreement with the coarser grid simulations. The dashed black line represents the laminar flow and the solid blue line turbulent experimental data [26] approximated by ${\Lambda }^{-0.5}=2{\text{log}}_{10}\left[2.25{\text{Re}}_b{\Lambda }^{0.5}\right]$. The two dotted vertical lines at ${\text{Re}}_b=865$ and 1077 correspond to the transitional limit according to Biau and Bottaro [8] (blue diamond) and Uhlmann et al. [7] (black empty circle; see also the three additional points shown with empty circles of the same authors for larger ${\text{Re}}_b$). The red asterisk (∗) at ${\text{Re}}_b=2205$ is the result from Gavrilakis [2]. To see the effect of periodicity, pieces of the curves for $\alpha =1.5$ and 2.0 have been added (in green, respectively, with dash-dotted and dashed lines).