Waviness and self-sustained turbulence in plane Couette-Poiseuille flow

Abstract

Direct numerical simulations of a Couette Poiseuille flow were performed near the transition to turbulence to investigate the nonlinear relationship between streak waviness and rolls. This relationship is a key step in Waleffe's model for a self sustaining process (SSP). Simulations were conducted for Reynolds numbers ranging from 500 to 940, and a range of initial perturbation amplitudes was used. In these simulations, the streaks, rolls, and streak waviness initially grow. The optimal time for this growth closely matches the linear transient growth period for small perturbations, but is much shorter when the initial perturbations are large and highly nonlinear. For higher Reynolds numbers and large initial perturbations, the velocity field reaches a turbulent steady state, while in the remaining cases the flow relaminarizes. The main result is that the waviness of the streaks is a quadratic function of the rolls, provided that the roll amplitude is sufficiently large.

Figures (12)

• Figure 1: (a,b) Schematic representation of a typical Couette-Poiseuille laminar flow considered. (c) Representative 3D snapshots of the streamwise velocity field $U_x$ (blue is low speed, red is high speed) for three cases, a decaying laminar flow at $Re=625$, a transitioning flow at $Re=806$, and a fully turbulent case at $Re=940$, from top to bottom, respectively. Rendering using the software VAPOR li2019vaporsgpearse_2023_vapor.
• Figure 2: (a,b) Snapshots of a typical flow field at $Re=806$ (Simulation T16 in Table \ref{['table1']} from Section \ref{['Numerical set up']}) at 5 different times $t0=0$, $t1=4$, $t2=7$, $t3=11$, $t4=35$ and $t5=100$, with time nondimensionalized by $L_0/U_0$. In (a) $xz$-plane at fixed $y = L_y/2$, and in (b) the $yz$-plane at fixed $x = L_x/2$ are presented. The heatmap in the two panels represent the fluctuations of the streamwise velocity component $u_x$, and the streamplot in panel (b) corresponds with the cross-stream flow fields.
• Figure 3: Streak amplitude $\langle |u_x| \rangle$ evolution as a function of time of the simulation T16 (see Table \ref{['table1']}). The times corresponding to the snapshots in Fig. \ref{['Fig2_streaks']} (a) and (b) are indicated along the curve.
• Figure 4: In the upper panels, the streamwise velocity field $u_x$ (left) and its decomposition into the straight part (middle) and wavy part (right) of a simulation with $Re=806$ (run T16), shown at fixed $y = L_y/2$, are displayed. Additionally, the power spectra of each field are presented in the lower panels.
• Figure 5: Temporal evolution of $\langle|u_x|\rangle$, $\langle|u_y|\rangle$, and $\langle|\omega_y^{wavy}|\rangle$ for run NL4 ($Re = 625$). The vertical axis is shown in logarithmic scale to highlight the exponential laminar decay, indicated by the three grey dash-dotted lines.