Turbulent and inertial roll waves in inclined film flow

Abstract

Conditions for the onset of high Reynolds number roll waves on inclined interfaces are sought. The model equations of Dressler [Commun. Pure Appl. Math. 2, 149 (1949)] and Needham and Merkin [Proc. R. Soc. London Ser. A 394, 259 (1984)] are analyzed using dynamic singularity theory (normal form techniques) and numerical methods. A new family of roll-wave solutions is discovered. They provide bounds and averages of the velocities of all roll waves at a given Froude number. These are favorably compared to the data of Brock [Proc. Am. Soc. Civ. Eng. 12, 2565 (1970)] and Brauner and Maron [Int. J. Heat Mass Transfer 25, 99 (1982)]. The average wave velocity is demonstrated to be approximately 1.5 times the average fluid velocity.

Weak Nonlinear Long Waves in Channel Flow with Internal Dissipation

“…Needham and Merkin (1984) [2] then use an averaging technique to calculate a one-parameter family (characterized by the propagation speed) of continuous traveling waves. These results are independently confirmed by Hwang and Chang (1987) [7] using a normal mode technique. However, the evolution of arbitrary initial disturbances and their relation to the calculated traveling waves is not considered in these works.…”

“…This term was included in the analysis by Needham and Merkin (1984) [2] to model internal dissipation empirically. We note that the dimensionless variables in (1) differ from those in equations (6)- (7) of Needham and Merkin (1984) [2]. To reconcile the notation, we must replace F R t, and σ in the latter by F 1/2 R/F 1/2 t/F 1/2 , and F, respectively.…”

Competition between kinematic and dynamic waves in floods on steep slopes

“…Two main approaches can be followed to establish the stability criteria of the plane-parallel flow: first, the hydraulic one, which finds long travelling-wave solutions for Froude numbers above 2 by means of the shallow-water equations (e.g. Jeffreys 1925;Dressler 1949;Needham & Merkin 1984;Hwang & Chang 1987;Yu & Kevorkian 1992;Chang, Demekhin & Kalaidin 2000); second, on the basis of the linearized Reynolds equations for a turbulent flow, with critical Froude numbers varying between 1.1 and 1.4 as the Reynolds number is varied over the realistic range of values 2 × 10 3 -10 5 (Demekhin, Kalaǐdin & Shapar' 2005). The effect of bottom topography on the stability of a turbulent flow over uneven surfaces was recently explored by Balmforth & Mandre (2004) following the hydraulic approach, who found that low-amplitude topography destabilizes turbulent roll waves and lowers the critical value of the Froude number required for instability.…”

Accelerated disturbance damping of an unknown distributed system by nonlinear feedback

“…We eschew the detailed Lie algebra involved for transforming Eq. 22 and refer the readers to our earlier derivation (Hwang and Chang, 1987) where y3, like Eq. 20a, is also a real coefficient given by Here, the amplitude r 2 ( f ) can again be extracted from an instantaneous measurement of z ( x , t ) by:…”