Viscous Instabilities in Transversely Strained Channel Flows

Abstract

We investigate here linear stability in a canonical three-dimensional boundary layer generated by the superposition of a spanwise pressure gradient upon an otherwise standard channel flow. As the main result, we introduce a simple coordinate transformation that enables the complete description of modal and non-modal stability using previous results on Poiseuille flow. We leverage this insight to derive closed forms for some relevant stability metrics. In particular, the critical Reynolds number for exponential-in-time growth is found to monotonically decrease with the strength of the cross-flow. A suitably chosen re-scaling, however, shows that the stability characteristics ultimately approach those of channel flow, despite the presence of a non-zero spanwise shear. Unstable eigenmodes akin to the Tollmien-Schlichting wave are found to propagate along the direction of the net flow. From a non-modal perspective, the maximal transient (algebraic) growth increases quadratically with the spanwise pressure differential and, similar to two-dimensional flows, is fueled by the lift-up effect. In this regard, the linear energy budget highlights a dramatic increase in energy production against the spanwise shear.

Figures (5)

• Figure 1: A sketch of the present flow geometry. Respectively, $\mathcal{G}_x$ and $\mathcal{G}_z$ are the streamwise and cross-stream pressure gradients.
• Figure 2: The critical parameters for eigenvalue instability in TSC flows plotted against $\Pi$; $(a)$, the critical Reynolds number -- the dashed line represents the re-scaled value involving the center-line velocity for the spanwise profile; $(b)$, the critical streamwise and spanwise wavenumbers. In both plots, the markers denote the critical parameters obtained via numerical experiments.
• Figure 3: $(a)$, the critical (streamwise) phase speed plotted against $\Pi$; the inset shows the re-scaled spanwise phase speed $c_z$. Note that formally, both are real. Once again, the markers denote values obtained from a numerical sweep. Panels $(b)$ and $(c)$, respectively, depict amplitudes of the streamwise and spanwise components corresponding to the unstable eigenmode for TSC flows. In $(c)$, the dashed line denotes $u\left(\Pi\to 0\right)$.
• Figure 4: $(a)$, the wavenumber pair yielding $G_{\max}$ for various $\Pi$ -- note that we have chosen $\beta_m = 2.05$ here; $(b)$, the optimal amplification $G_{\max}$ normalized by its value for Poiseuille flow $\left(\Pi = 0\right)$; from Equations (\ref{['eqn:gmax_ppf']}) and (\ref{['eqn:gmax_tsc']}), this should roughly scale as $\sim 1+\Pi^2$. In either panel, the markers denote data from numerical experiments.
• Figure 5: $(a)$ and $(b)$, the initial condition and response field (at $t = t_{\max}$) associated with $G_{\max}$ at $\Pi = 2.5$ -- color/quivers denote the streamwise/cross-stream components of the disturbance; $(c)$: for select $\Pi$, the time evolution of the linear budget for the energy-optimal initial condition. In the main panel, the solid line denotes $\mathcal{P}_W$ and the dashed line $\varepsilon$.