Parametric resonance in unsteady watertable flow

Abstract

The stability of unsteady open-channel flow down an inclined plane is studied using an iterative approach based on the direct and adjoint stability equations combined with a physically justified energy measure. An efficient parametric resonance mechanism has been identified between the exogenous base-flow oscillations and the intrinsic frequencies of streamwise disturbance vortices. This resonance results in strong amplification over a substantial range of the governing parameters, favouring streamwise elongated structures. The optimal frequency for a maximal disturbance response can be efficiently approximated from simpler steady calculations; two frequency-selection criteria are given for this purpose. The analysis generalizes earlier work on steady watertable flow and provides an effective framework and starting point for further work on pattern formation in harmonically forced open-channel flows.

River bedform inception by flow unsteadiness: A modal and nonmodal analysis

“…The evaluation of the growth function by the direct-adjoint procedure for a wide range of time horizons is computationally demanding and usually unnecessary. All unsteady curves are instead obtained by the following simplified procedure [44]: (i) the growth functionĜ(t) is computed by the more efficient SVD method for the corresponding steady problem; (ii) the time t max at whichĜ(t) reaches a maximum is identified; (iii) the iterative direct-adjoint technique is applied for t = t max to determine the optimal initial condition q 0 ; (iv) the direct problem (19a) is integrated in time starting from the optimal initial condition and the growth function is evaluated.…”

Optimal disturbance growth in shear‐imposed falling film

“…Later, Olsson and Henningson showed the existence of transient disturbance growth for a watertable flow. Recently, the model of Olsson and Henningson was further investigated by Camporeale and Schmid to incorporate the influence of unsteady base flow on the amplification mechanism. In this article, our aim is to explore the effect of external shear stress on the disturbance energy growth found in the free surface flow, or equivalently, the role of base flow in the disturbance energy growth for a free surface flow.…”

“…In general, the present study of free surface flow is a fascinating topic because it provides both surface and shear modes in comparison with the wall-bounded shear flow where only the shear mode is observed. In this context, Gustavsson 50 to incorporate the influence of unsteady base flow on the amplification mechanism. In this article, our aim is to explore the effect of external shear stress on the disturbance energy growth found in the free surface flow, or equivalently, the role of base flow in the disturbance energy growth for a free surface flow.…”