Large-scale streak instabilities of transitional channel flow

Abstract

The emergence of large-scale spatial modulations of turbulent channel flow, as the Reynolds number is decreased, is addressed numerically using the framework of linear stability analysis. Such modulations are known as the precursors of laminar-turbulent patterns found near the onset of relaminarisation. A synthetic two-dimensional base flow is constructed by adding finite-amplitude streaks to the turbulent mean flow. The streak mode is chosen as the leading resolvent mode from linear response theory. Besides, turbulent fluctuations can be taken into account or not by using a simple Cess eddy viscosity model. The linear stability of the base flow is considered by searching for unstable eigenmodes with wavelengths larger than the base flow streaks. As the streak amplitude is increased in the presence of the turbulent closure, the base flow loses its stability to a large-scale modulation below a critical Reynolds number value. The structure of the corresponding eigenmode, its critical Reynolds number, its critical angle and wavelengths are all fully consistent with the onset of turbulent modulations from the literature. The existence of a threshold value of the Reynolds number is directly related to the presence of an eddy viscosity, and is justified using an energy budget. The values of the critical streak amplitudes are discussed in relation with those relevant to turbulent flows.

Figures (20)

• Figure 1: Contours of streamwise velocity fluctuations in the plane $y^+\approx 35$ from a DNS at (a) $Re_{\tau} = 98$, (b) $Re_{\tau} = 92$ and (b) $Re_{\tau} = 71$ ($y\approx0.36$ for $Re_{\tau}=98,92$ and $y\approx0.49$ for $Re_{\tau}=71$). The black vertical segment at the bottom left corner of each panel indicates a spanwise length of $1000$ wall units.
• Figure 2: Optimal harmonic forcing (a-b-c-d) and response velocity field (e-f-g-h) obtained from the resolvent analysis of the mean flow for (a-b-e-f) $Re_{\tau}=71$ and (c-d-g-h) $Re_{\tau}=105$. In all panels contours denote the streamwise component while quivers stand for transverse components. The scale of the arrows in the top row is ten times larger than in the bottom row. Streamwise uniform case ($k_x=0$). (a-c-e-g) Quasi-laminar model; (b-d-f-h) eddy viscosity model.
• Figure 3: Eigenvalues for the streak stability problem (only a subset of the computed spectra is shown) for $Re_{\tau}=71$ and $k_x=0.18$. The eigenvalues ($\bullet$ for stable modes, $\star$ for unstable modes) are coloured with the corresponding root of unity factor $\gamma = j/N_u$ for $j=0,\dots,N_u/2$ (the eigenvalues for $\gamma \in (0.5,1.0)$ are equal to those for $\gamma \in (0,0.5)$ and the corresponding modes are the same up to a reflection in the spanwise direction). (a-c-e) Quasi-laminar model; (b-d-f) eddy viscosity model. Streak amplitudes increase from top to bottom and are (a) $0.10$, (c) $0.16$ and (e) $0.20$ for the quasi-laminar model and (b) $0.30$, (d) $0.40$ and (f) $0.50$ for the eddy viscosity model.
• Figure 4: Variation of the leading eigenvalues with $Re$ and streak amplitude $A_s$ for $k_x=0.18$ ($\bullet$ for stable modes, $\star$ for unstable modes). (a-b): growth rate. (c-d): phase velocity. (a-c) Quasi-laminar model; (b-d) eddy viscosity model.
• Figure 5: Eigenvalues coloured by the large-scale spanwise energy ratio $r_{LS}$ (only a subset of the computed spectra is shown) for $Re_{\tau}=71$ and $k_x=0.18$ ($\bullet$ for stable modes, $\star$ for unstable modes). (a) Quasi-laminar model and $A_s=0.2$; (b) eddy viscosity model and $A_s=0.5$.