Stability of roll waves in open channel flows

Abstract

The stability of finite amplitude roll waves that may develop at a liquid free surface in inclined open channels of arbitrary cross-section is studied. In the framework of shallow water theory with turbulent friction the modulation equations for wave series are derived and a nonlinear stability criterion is obtained.

Spectral Stability of Inviscid Roll Waves

“…We see clearly that (1.3) does not correctly predict low-frequency spectral behavior nor long-time dynamical stability of roll waves of any kind. Indeed, it gives both false positive and false negative predictions of spectral stability, invalidating in a strong sense the model (1.3) proposed in [BL02]. This resolution, along with the analytical and numerical validation of the inviscid Whitham system (1.2) , demonstrates the enormous benefit of working with the exact eigenvalue equations (1.9) derived from sound mathematical bases.…”

“…In Figure 4(a), we compare the above stability results to the low-frequency stability predictions of model (1.3), given in terms of average relative height H/H s as in [BL02], where here H denotes the average of the height profile H over a period. Here, the green curves are the stability boundaries of [BL02] and the red and blue boundaries the stability boundaries of Figure 3. We see clearly that (1.3) does not correctly predict low-frequency spectral behavior nor long-time dynamical stability of roll waves of any kind.…”

Roll waves and plugs in two-layer flows

“…Throughout the paper, the averaged equations for a separated-flow model of instationary stratified flow regimes with turbulent friction on the wall and at the interface are considered. The nonlinear stability problem of roll waves for flows in inclined open channel of arbitrary shape is solved by deriving the modulation equations [5]. periodic patterns of bores separated by continuous profiles of the interface, and plug flow regimes for zero-mean pressure gradient gas-liquid flow system, which consist of successive cells made of liquid, separated from one another by a liquid layer topped by a gaseous bubble.…”

Modulation Equations for Roll Waves on Vertically Falling Films of a Power-Law Fluid

“…A report on roll waves can be found in extensive references quoted by Ng & Mei [1]. The roll waves which occur in inclined open channels are important in drainage problems and have received an extensive treatement in turbulent regime by Jefffreys [2], Dressler [3], Boudlal & Liapidevskii [4], among others, and for a laminar sheet flow by considering the flow with quadratic distribution of velocity profile (Alekseenko & Nakoryakov [5]; Buchin & Shaposhnikova [6]; Julien & Hartley [7]; Boudlal & Liapidevski [8]).…”