Three-dimensional disturbances in channel flows

Abstract

In this paper we conduct a linear stability analysis of three-dimensional two-fluid flows and use an energy method to comment on its stability. The governing equations are solved using a Chebyshev-tau D 2 method that reduces the order of the coupled governing Orr-Sommerfeld and Squire equation and hence achieves more accurate results. A new norm, called the M-norm, is defined to overcome the problem of nonconvergence of the disturbance energy. The maximum amplification of O͑10 3 ͒ is achieved for streamwise independent disturbances due to the "lift-up effect," as is the case of three-dimensional single-fluid flow. In contrast to two-dimensional flows, where the adjoint of the leading mode influences the growth, the three-dimensional single-fluid flow growth is influenced by the adjoint of the second mode. Although most growth in three-dimensional two-fluid flow is due to the contribution of the adjoint of the second mode, at large time the interfacial mode contributes to most growth.

Disturbance growth in two-fluid channel flow: The role of capillarity

“…South and Hooper (1999), introducing a similar ''M-norm," found that values of this coefficient much smaller than unity work well and that this coefficient ideally should include an (m À 1) factor to ensure that the single fluid result is recovered for the limit m ? 1 (see also Malik and Hooper, 2007). In this work, the limit of zero interfacial tension is not considered, although convergence is explicitly re-checked whenever k 2 /We ( 1.…”

Non-modal stability of round viscous jets

“…The usual choice for the norm in studies of transient growth in single-fluid flows is the kinetic energy of the disturbance confined to single periods in θ and t (Butler & Farrel 1992;Yecko & Zaleski 2005). When studying the transient growth in two-fluid flows it is necessary to include an additional term related to the interfacial energy into the expression for the norm, otherwise the calculations converge very poorly (Renardy 1987;van Noorden et al 1998;South & Hooper 1999;Malik & Hooper 2007). We consider the following expression for the energy norm:…”

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

“…This approach has elucidated the role played by linear mechanisms in the triggering of instabilities in simple shear flows (e.g., [27][28][29] In contrast, in a morphodynamic context this approach has only recently been applied, where it demonstrated the potential for transient growth for one-dimensional bed waves [30], river dunes [31], and bars [32]. Moreover, the transient dynamics of river patterns has not been investigated in the presence of flow variations.…”