Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the Orr- Somrnerfeld and Squire （OSS） equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a re- orthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes （N-S） equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is pro- posed. The algorithm is then applied to a one-dimensional base flow （the plane Poiseuille flow） and a two-dimensional base flow （the plane Poiseuille flow with a low-speed streak） in a channel. For one-dimensional cases, the effects of the spanwise width of the chan- nel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is character- ized by high- and low-speed streaks and the corresponding streamwise vortical structures. The lift-up mechanism that induces the transient growth of perturbations is discussed. The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one- and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the cross- check method.