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Stability of an unsteady channel flow is investigated by incorporating the effect of the base flow change with time and comparing with the optimal growth of the normal mode analysis. The existing literature shows that the modal analysis can be done on the velocity profiles in a quasi-steady manner to study the stability characteristics of unsteady internal flows. But, in the current study, it is shown that the mode obtained from the optimal growth analysis provides higher growth and estimates the structures observed in experiments. Initially, the optimal mode is obtained using Farrell’s analysis, and its energy growth is compared with that of the most unstable eigenmode. Later, to incorporate the base flow change, the optimal mode for maximum growth is obtained by solving the problem with a variational approach. The results are compared to provide justification of Farrell’s analysis, which is quasi-steady, against the requirement of complete study considering the time dependent base flow.

Summary This paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time‐based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time‐based numerical integration of the linearized Navier‐Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity‐radial vorticity formulation. Even number of Chebyshev‐Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity.

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