Extended Squire’s transformation and its consequences for transient growth in a confined shear flow

Abstract

AbstractThe classical Squire transformation is extended to the entire eigenfunction structure of both Orr–Sommerfeld and Squire modes. For arbitrary Reynolds numbers $\mathit{Re}$, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number $\mathit{Re}_{2D}$. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal gain at all optimization times $t$ with ${t}/{\mathit{Re}}$ finite or large. This result is an extension of the well-known scaling laws first obtained by Gustavsson (J. Fluid Mech., vol. 224, 1991, pp. 241–260) and Reddy & Henningson (J. Fluid Mech., vol. 252, 1993, pp. 209–238) for arbitrary $\alpha \mathit{Re}$, where $\alpha $ is the streamwise wavenumber. The Squire transformation is also extended to the adjoint problem and, hence, the adjoint Orr–Sommerfeld and Squire modes. It is, thus, demonstrated that the long-time optimal growth of 3D perturbations as given by the exponential growth (or decay) of the leading eigenmode times an extra gain representing its receptivity, may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms equal to $1+ \left (\beta \mathit{Re}/\mathit{Re}_{2D}\right )^2 \mathcal{G}$, where $\beta $ is the spanwise wavenumber and $\mathcal{G}$ is a known expression. For example, when the leading eigenmode is an Orr–Sommerfeld mode, it is given by the product of respective gains from the 2D Orr mechanism and an analytical expression representing the 3D lift-up mechanism. Whereas if the leading eigenmode is a Squire mode, the extra gain is shown to be solely due to the 3D lift-up mechanism. Direct numerical solutions of the optimal gain for plane Poiseuille and plane Couette flow confirm the novel predictions of the Squire transformation extended to the initial-value problem. These results are also extended to confined shear flows in the presence of a temperature gradient.