Abstract
Linear and weakly nonlinear instability of shallow mixing layers is analysed in the present paper. It is assumed that the resistance force varies in the transverse direction. Linear stability problem is solved numerically using collocation method. It is shown that the increase in the ratio of the friction coefficients in the main channel to that in the floodplain has a stabilizing influence on the flow. The amplitude evolution equation for the most unstable mode (the complex Ginzburg–Landau equation) is derived from the shallow water equations under the rigid-lid assumption. Results of numerical calculations are presented.
Highlights
Understanding the interaction between fast and slow fluid streams at river junctions and in compound channels is important for a proper description of mass and momentum transfer in shallow flows
In order to analyse the problem, hydraulic engineers apply depth-averaged shallow water equations [1] where either Chezy or Manning formulas are used to take into account the effect of bottom friction. ese formulas contain a constant friction coefficient determined from empirical relationships [1] if the Reynolds number of the flow and surface roughness are given. ere are cases, where the resistance force changes considerably in the transverse direction [2, 3]
Numerical Results e complex Ginzburg-Landau equation as the amplitude evolution equation for the most unstable mode is analysed in the previous section using asymptotic expansion (8)
Summary
Understanding the interaction between fast and slow fluid streams at river junctions and in compound channels is important for a proper description of mass and momentum transfer in shallow flows. Ere are cases, where the resistance force changes considerably in the transverse direction [2, 3]. One example of such a situation is flow in compound channels or rivers in case of floods. Us, it is necessary to analyse factors affecting shallow flow instability and development of perturbations in a weakly nonlinear regime. Ree different approaches for the investigation of shallow water flows are suggested in [4]: experimental studies, numerical modelling, and stability analysis. E development of perturbations can be analysed from a spatial point of view [13,14,15,16]. Calculations show that bed friction reduces the spatial growth rate of perturbations
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