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We present the possibility of energy transfer and budget analysis fortransient flow using eigenmodes of the operator from the Navier-Stokesequation. We derive the energy transfer equation, which provides the energybudget for the eigenmodes, through the Galerkin projection of the equationusing the bi-orthogonality of the eigenmodes and the adjoint mode. Energybudget and transfer analysis between modes were conducted for two-dimensionalflow around a cylinder with eigenmodes growing or decaying from a steady flow.Using the linearized energy transfer equation and eigenmodes from globalstability analysis, we identify the energy budget and spatial distribution thatdetermine mode growth rates. Moreover, energy budget and transfer analysis arerealized by considering the time evolution of the eigenmodes, even during thenonlinear development of the eigenmodes. By introducing time-varying dynamicmode decomposition with a phase-control strategy for multiple time-seriesdatasets from numerical simulations of the phase-controlled initial flow,time-varying eigenmodes are extracted in the transient two-dimensional cylinderflow. With the extracted time-dependent modes and the derived energy transferequations, the time evolution of the energy budget and spatial distribution ofenergy transfer can be computed until the eigenmodes developed from the steadyfield reach the post-transient periodic flow. From the time variation of theenergy budget and the transfer distribution, the transient process of thecylinder flow can be characterized by the ratio of the magnitude of viscousdiffusion for the eigenmode and energy transfer from the base flow.

Energy transfer and budget analysis for transient process with operator-driven reduced-order model