Applied Mathematics and Mechanics >
Algorithm for transient growth of perturbations in channel Poiseuille flow
Received date: 2016-08-23
Revised date: 2017-05-15
Online published: 2017-11-01
Supported by
Project supported by the National Natural Science Foundation of China (No. 11372140)
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-Sommerfeld and Squire (OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a reorthogonalization 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 proposed. 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 channel 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 characterized 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 crosscheck method.
Jianlei ZHANG , Gang DONG , Yi LI . Algorithm for transient growth of perturbations in channel Poiseuille flow[J]. Applied Mathematics and Mechanics, 2017 , 38(11) : 1635 -1650 . DOI: 10.1007/s10483-017-2275-9
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