Transient growth of optimal perturbation in a decaying channel flow

Abstract

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.

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

“…The deceleration phase of the trapezoidal piston motion and the flow thereafter generated velocity profiles similar to those observed for a suddenly blocked duct flow; as shown in Das & Arakeri (1998, figure 10) and Weinbaum & Parker (1975, figure 7). Here, the critical flow characteristics such as the reverse-flow region near the wall, inflectional velocity profiles and post-deceleration diffusion-based-flow development were some of the similarities found between the two kind of flows (Nayak & Das 2017). Thus, the deceleration phase of the flow and the gradual flow development owing to viscous diffusion, after the piston has stopped, emulate the phenomena of impulsively blocked flow to a certain extent.…”

“…For a suddenly blocked channel, the condition leads to a limit on Reynolds number with as discussed in Hall & Parker (1976). Though it advocates the potential of the perturbation growth, it does not necessarily ensure the presence of instability owing to the decaying nature of the flow (Nayak & Das 2017). Stability characteristics of the sort of flow, currently under consideration, have been explained through quasi-steady normal-mode approaches in Das & Arakeri (1998) and Ghidaoui & Kolyshkin (2002).…”

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

“…It is shown that the optimal mode can be obtained by using an iterative technique with numerical integration of state and adjoint equations. It may not seem useful for steady pipe‐Poiseuille flow but has more effectiveness while studying the time‐dependent flows …”

“…Therefore, the functional G ( t ), defined on the space of state vector q as needs to be maximized against the spatial distribution of initial perturbation q (0). Details about the procedure of variational formulation and the iterative techniques with adjoint operators that can be used to obtain the optimal initial perturbation are comprehensively discussed in the works of Nayak and Das, Schmid, Zhuravlev and Razdoburdin . Thus, the adjoint equations in terms of the adjoint variables corresponding to the dynamical system represented by are provided excluding the rigorous derivation.…”

“…It may not seem useful for steady pipe-Poiseuille flow but has more effectiveness while studying the time-dependent flows. 39 With the perturbation state vector defined as q = [Φ Ω] T , the aim is to determine an optimal initial perturbation q(0) such that the energy growth G(t) is maximized over a time interval t. Therefore, the functional G(t), defined on the space…”