A pseudospectral approach applicable for time integration of linearized N‐S operator that removes pole singularity and physically spurious eigenmodes

Abstract

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.

Experimental and numerical investigation of flow instability in a transient pipe flow

“…In the current context, the state vector is the perturbation quantity represented by , where and are the transformed radial velocity and radial vorticity, respectively, as (Burridge & Drazin 1969; Nayak & Das 2019) where , and are the perturbation velocities and represents the radial coordinate. Here, and represent the streamwise wavenumber and discrete azimuthal wavenumber, respectively, with .…”

“…Here, and represent the streamwise wavenumber and discrete azimuthal wavenumber, respectively, with . The energy norm is defined with the kinetic energy density as Details about the form of perturbation state vector , the governing perturbation equations and the solution technique are presented in Nayak & Das (2019).…”