Research Article Linear and Weakly Nonlinear Instability of Shallow Mixing Layers with Variable Friction Irina Eglite and Andrei Kolyshkin Department of Engineering Mathematics, Riga Technical University, Daugavgrivas Street 2, Riga LV-1007, Latvia Correspondence should be addressed to Andrei Kolyshkin; andrejs.koliskins@rbs.lv Received 23 August 2017; Revised 30 October 2017; Accepted 27 November 2017; Published 19 March 2018 Academic Editor: Giuseppe Oliveto Copyright © 2018 Irina Eglite and Andrei Kolyshkin. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. e 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. 1. Introduction 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 Ch´ ezy 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, however, 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. Hence, the analysis of instability characteristics of shallow mixing layers with variable friction is important from environmental point of view. In particular, contami- nants and residues can accumulate in some parts of the flow due to instability. us, it is necessary to analyse factors affecting shallow flow instability and development of per- turbations in a weakly nonlinear regime. ree different approaches for the investigation of shallow water flows are suggested in [4]: experimental studies, nu- merical modelling, and stability analysis. Experimental data [5–7] indicate that bottom friction has a stabilizing influence on the flow. Temporal linear stability analysis of shallow mixing layers [8–12] shows that bed friction reduces the width of a mixing layer and stabilizes the flow. e development of perturbations can also be analysed from a spatial point of view [13–16]. Calculations show that bed friction reduces the spatial growth rate of perturbations. ere are other factors affecting stability characteristics of shallow mixing layers: (a) flow curvature, (b) presence of solid particles in a fluid stream, and (c) variable friction force in the transverse direction. ese aspects of the linear sta- bility problem are analysed in [13–16]. In particular, a stably curved mixing layer has a stabilizing influence on the flow, while an unstably curved mixing layer destabilises the flow. In addition, the presence of solid particles in the stream reduces the growth rate of perturbations. Linear stability analysis is a useful tool for calculating critical values of the parameters characterizing the flow. However, linear theory cannot be used to analyse the de- velopment of perturbations in unstable regime. Assuming that the bed friction number is slightly smaller than the critical value (i.e., the flow is linearly unstable with a small growth rate), weakly nonlinear theories can be used to obtain an amplitude evolution equation for the most unstable mode [17–22]. e analysis in [17–21] shows that the amplitude Hindawi Advances in Civil Engineering Volume 2018, Article ID 8079647, 10 pages https://doi.org/10.1155/2018/8079647