Advances in Civil Engineering

Volume 2018, Article ID 8079647, 10 pages

https://doi.org/10.1155/2018/8079647

## Linear and Weakly Nonlinear Instability of Shallow Mixing Layers with Variable Friction

Correspondence should be addressed to Andrei Kolyshkin; vl.sbr@sniksilok.sjerdna

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. This 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.

#### 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.

#### 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 Chézy or Manning formulas are used to take into account the effect of bottom friction. These formulas contain a constant friction coefficient determined from empirical relationships [1] if the Reynolds number of the flow and surface roughness are given. There 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, contaminants and residues can accumulate in some parts of the flow due to instability. Thus, it is necessary to analyse factors affecting shallow flow instability and development of perturbations in a weakly nonlinear regime.

Three different approaches for the investigation of shallow water flows are suggested in [4]: experimental studies, numerical 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. The 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.

There 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. These aspects of the linear stability 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 development 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]. The analysis in [17–21] shows that the amplitude evolution equation is the complex Ginzburg–Landau equation (GLE) under the rigid-lid assumption (free surface is treated as a rigid lid). As it is shown in [10], rigid-lid assumption works well for small Froude numbers. The Froude number is of order 0.2-0.3 in experiments [2] so that rigid-lid assumption is justified. The following cases of shallow flows are considered in [17–21]: (a) Rayleigh friction [19], (b) bottom friction modelled by the Chézy formula [21], and (c) generalization of the analysis in [21] for the case of slightly curved mixing layers and two-phase flows in [17–19]. Recent application of weakly nonlinear theory to shallow water flows without rigid-lid assumption [22] shows that the amplitude evolution equation for the most unstable mode is the complex Schrödinger equation. Since the solutions of the GLE vary from truly deterministic to almost chaotic [23] (depending on the values of the coefficients), in many cases, the GLE is used as a phenomenological equation for the analysis of spatiotemporal dynamics of complex flows. The coefficients of the equation are estimated using experimental data. Complex phenomena in fluid mechanics (see, e.g., [24]) can be modelled using the obtained equation.

In the present paper, we perform linear and weakly nonlinear analysis of shallow mixing layers under the assumption that the friction force changes in the transverse direction. Weakly nonlinear analysis shows that the amplitude evolution equation for the most unstable mode is the complex GLE. The equation is derived from the shallow water equations under the rigid-lid assumption. The coefficients of the equation are calculated in closed form using linear stability characteristics of the flow. Results of numerical calculations are presented. Practical applications of the proposed model are also discussed.

#### 2. Linear Stability Problem

Consider the system of shallow water equations under the rigid-lid assumption written in the form [25]where and are the depth-averaged velocity components in the and directions, respectively, is water depth, is the pressure, and is the friction coefficient depending on the transverse coordinate . The function is assumed to be of the formwhere is a constant and is an arbitrary differentiable “shape” function.

Since friction terms in (2)-(3) do not contain derivatives, the boundary conditions for the longitudinal velocity component are not needed (otherwise the problem would be overdetermined). Thus, the boundary conditions for the velocity components are essentially the same as in the case of inviscid flow (we need only zero normal velocity at the boundaries).

The boundary conditions are

The assumption of an unbounded layer in the transverse direction is widely used in the stability analysis of mixing layers and wakes (see, e.g., [26, 27]) and is adopted here.

Introducing the stream function by the relationsand eliminating the pressure from (1)–(3), we obtainwhere is the derivative of with respect to and .

Consider a perturbed solution to (7) of the formwhere is the base flow velocity. The role of the small parameter will be clarified later.

Substituting (8) into (7) and linearizing the resulting equation in the neighbourhood of the base flow , we obtainwhere

Using the method of normal modes, we represent the function in the formwhere is the amplitude of the normal perturbation and is the wavenumber.

Substituting (11) into (9), we obtainwhere is the bed friction number and is the half-width of the mixing layer.

The boundary conditions follow from (5) and have the form

Complex eigenvalues determine the linear stability of the base flow. The flow is linearly stable if all and unstable if at least one . Problems (12) and (13) are solved numerically by means of the collocation method based on the Chebyshev polynomials [15]. In the classical hydrodynamic stability theory (see, e.g., [26, 27]), the base flow is obtained as a simple one-dimensional steady solution of the equations of motion (e.g., a plane Poiseuille flow is obtained as a steady one-dimensional solution of the Navier–Stokes equations). It is impossible to find a simple analytical solution of the form for shallow water equations (1)–(3) due to the empirical character of the Chézy formula in (2) and (3). The paper [28] gives an example of a longitudinal velocity distribution in the form of a generalized power law. This profile does not have an inflection point (recall that the presence of an inflection point is the necessary condition for instability of inviscid flows [26]). The obtained formula for the longitudinal dispersion coefficient in [28] is based on the assumption of straight symmetrical channel and uniform flow. The resulting formula is multiplied by a “catch-all” revision coefficient taking into account many kinds of nonuniformities in natural rivers. However, this adjustment does not change the shape of the longitudinal velocity distribution. The mixing layer case (the case considered in our paper) has a completely different longitudinal velocity distribution [2]. The presence of a porous layer with much higher resistance than in the main channel results in a highly nonuniform velocity distribution resembling a hyperbolic tangent function with some asymmetry (however, the asymmetric profile is obtained after all the instabilities have been taken into account). The model profile has an inflection point. Thus, instabilities can occur. The hyperbolic tangent profile is widely used in stability analyses of shallow flows [8–12]. The goal of our investigation is to analyse instabilities of the flow, obtain the linear stability characteristics, and consider the development of perturbations in a weakly nonlinear regime.

Assuming that the velocity of undisturbed flow is and at and , respectively, the function is chosen in the formwhere all the dimensional quantities are represented with tildes. Using as the velocity scale and as the length scale, we rewrite (14) in the dimensionless formwhere is the velocity ratio. All calculations in the paper are performed for the case . The base flow velocity distribution is given in Figure 1. The domain of the flow is defined as follows: the transverse coordinate varies from to . A porous layer occupies the region , while the region corresponds to the main channel. Away from the shear layer, the base flow velocity approaches the undisturbed values and , respectively. We assume that the volume fraction is small (in such a case, as it is shown in [2] shallow water equations of the form (1)–(3) can be used to analyse the problem).