Using modal decompositions to explain the sudden expansion of the mixing layer in the wake of a groyne in a shallow ﬂow

The sudden expansion of the mixing layer created in the wake of a single groyne is investigated using Particle Image Velocimetry (PIV). In the region of the sudden expansion a patch of high Reynolds shear stresses are observed. Using low-order representations, created from a Dynamic Mode Decomposition and a search criteria based on a Proper Orthogonal Decomposition, the spatio-temporal mechanism of the sudden expansion is investigated. The present study demonstrates the sudden expansion is created by the periodic merging of eddies. These eddies originate from the upstream separation and the tip of the groyne and merge with recirculating eddies created, downstream of the groyne, at the interface of the mixing layer and the lateral wall.


Introduction
In natural flows such as rivers and estuaries, groynes are installed to prevent bank scouring [34,45] or to create and enhance fish habitats [15]. The majority of these natural flows are bounded flows in a domain for which two dimensions, namely that in the direction of the flow, as well as one traverse di-mension, greatly exceed the third dimension, consequently they fulfil definition of a shallow flow [18].
Dependent upon the magnitude of the transversal velocity gradient, a topographical obstruction of any flow can lead to the formation of a mixing layer.
In contrast to in deep flows, the large-scale coherent turbulent structures which 10 populate the far field of a shallow mixing layer, can almost extend the whole depth of the flow. As a consequence these quasi-two-dimensional coherent structures (Q2CS), and their spatio-temporal dynamics, are easily influenced by bedfriction [30,51,10,44]. From an environmental perspective these Q2CS are of great significance, as their spatio-temporal behaviour governs mass and momen- 15 tum exchange. Some examples of this can be found in their key role to predict the concentration of pollutants, nutrients and the rates of sediment transport [36,5,46,8].
Many previous works have investigated the spatio-temporal dynamics of Q2CS created by plane shear instabilities e.g. [10,51,36], and have shown that 20 due to the effects of bed-friction the spread/growth rate of a shallow mixing layer is modified. In an experimental study to investigate the effect of topographical forcing on a shallow flow, Talstra et al. [49] found that unlike in a deep flow, [2], the shallow flow mixing layer bound a second counter rotating recirculation cell.
They also found at the downstream edge of the first recirculation cell there was 25 a sudden expansion in the mixing layer. In Talstra [48] it is suggested that this sudden expansion is caused by the merging of vortex shed from the tip of the obstacle and those associated to the largest downstream gyre. In a more recent study, this sudden expansion was also found to occur when a shallow flow was obstructed by a single lateral groyne, it was also observed that the length of 30 reattachment of the mixing layer with the wall was protracted compared with the a deep case [39,40]. A number of time-averaged experimental studies have previously investigated this case, but have neither observed or explained this phenomena [12,1,20,13]. The Reynolds Averaged Navier-Stokes (RANS k-ω) simulations of Chrisohoides et al. [9] observed the dual cell system and found 35 that it was stable but periodically horizontally expanding and contracting. They 2 further found that eddies from upstream and shed from the tip of the obstacle were engulfed by the second downstream recirculation cell, and as the mixing layer reattached with the lateral wall, vorticity was injected back upstream; however they did not observe a sudden expansion the mixing layer. Safarzadeh 40 and Brevis [39] recently explained that due to the anisotropy associated to the flow system, RANS simulations, based on isotropic closure models, will not be able to simulate the expansion of the mixing layer or predict the length of reattachment. Other Computational Fluid Dynamics (CFD) studies relating to a single groyne have only focused on the turbulent mechanisms upstream and in 45 the near wake of the groyne e.g. [14,24,23,32,22], whilst not directly related to this study it is inferred that these complexities have implications on the dynamics of the mixing layer downstream. From all of these findings it is clear that the dynamics of a shallow mixing layer produced in the downstream wake of a single groyne are complex and governed by a number of factors. 50 The main goal of this research is to investigate the spatio-temporal mechanisms and Q2CS relating to the sudden expansion of the mixing layer. Whilst, the occurrence of this has been previously observed, the physics leading to it has not. Such an interpretation of the physics is important, as this understanding will help one to hypothesise how different flow and boundary conditions will 55 affect the formation and dynamics of the sudden expansion. This is particularly important from an environmental perspective as the increased moment fluxes relating to this phenomenon can lead to enhanced scouring / mixing processes.
To investigate these mechanisms an experimental Particle Image Velocimetry (PIV) study is undertaken. To describe the spatio-temporal mechanism a low-60 order reconstruction of the flow is made from the temporally orthogonal Dynamic Mode Decomposition (DMD) [41,37] using on a search criteria based on a spatially orthogonal Proper Orthogonal Decomposition (POD) [3,4].

Proper Orthogonal Decomposition
POD is a linear statistical method commonly used in fluid mechanics for the extraction and analysis of energy meaningful turbulent structures [3,4]. V(x, y; t), is considered, each of which is of dimension X × Y . The method requires the construction of an N × T matrix W from T columns, w(t) of length N = XY , each one corresponding to a column-vector version of a transformed snapshot V(x, y; t). A POD can be obtained by : where S is the singular valies matrix of dimension Θ × Θ, (Θ are the number of modes of the decomposition, and (·) * represents a conjugate transpose matrix operation). The λ = diag(S) 2 /(N − 1) is the vector containing the contribution to the total variance of each mode. The elements in λ are ordered in descending 5 rank order, i.e. (λ 1 ≥ λ 2 ≥ . . . λ Θ ≥ 0). In practical terms the matrix Φ of dimension N × Θ contains the spatial structure of each of the modes and the matrix C of dimension Θ × Θ contains the coefficients representing the time evolution of the modes. Furthermore, the percentage turbulent kinetic energy contribution, E, of a each POD mode can be obtained via: · 100 (2)

Dynamic Mode Decomposition
The Dynamic Mode Decomposition (DMD) algorithm was introduced into fluid mechanics by Schmid [41] & Rowley et al.
[37], based on a Arnoldi Eigenvalue algorithm suggested by Ruhe [38]. Unlike POD, which is based upon a co-variance matrix, using and Arnoldi aproximation the DMD fits a high-degree polynomial to a Krylov sequence of snapshots, which are assumed to become linearly dependent after a sufficient number of snapshots [29,43]. As discussed by Schmid [42] such a representation of a nonlinear process by a linear sampleto-sample map can be closely linked to the concept of a Koopman operator, an analysis tool for dynamical systems. For complex flow systems containing superpositions of turbulent structures and mechanisms, the DMD algorithm can be used to extract spatial modes with single 'pure' frequencies. There are currently a number of methods by which one can compute a DMD, and the method used in the present study is the popular SVD based method; this approach has been shown to be less susceptible to experimental noise [41]. The algorithm is outlined below, although the reader is directed to Schmid [41], Jovanović et al. [19], Tu et al. [50] for the full mathematical description. The DMD algorithm begins with a similar transformation as POD: where τ = (T − 1), and the super-scripts A and B denote the two W matrices of dimension N × τ . A SVD of W A is computed, such that: 6 whereΦ,S andC are the POD modes, the singular values and the temporal and its complex Eigenvalues, µ i , and Eigenvectors, z i , are computed where i = 1 . . . τ . At this point the method of Jovanović et al. [19] is used, as this creates a set of amplitudes for each spatial mode. Following Jovanović et al. [19] a Vandermonde matrix is created from the complex eigenvalues: where i = 1 . . . τ and j = 1 . . . τ , and the spatial modes are created by Ψ =ΦZ, where Z is a matrix of complex Eigenvectors previously computed. Furthermore, a matrix of amplitudes, D α , are created and the original input, W A , can be expressed as: where D α is of dimension τ × τ . Ψ, is a matrix of dimension N × τ containing the DMD spatial modes. α = diag(D α ) relates to their amplitude Ψ, but not their variance, and Q contains the coefficients representing the time evolution of the modes. In practical terms, as shown by [7] the angle between the real and imaginary part of, z i , can be used to describe the frequency relating to each Ψ and expressed as a Strouhal number by: where, i = 1 . . . τ , L is the length of the groyne and U 0 is the bulk velocity and 95 f 0 is the acquisition frequency.

Time-averaged statistics
As previously outlined, the emphasis of the present work is to describe the mechanisms underpinning the sudden expansion of the mixing layer. Before in-  Fig. 2(a) by a white circle). The sudden expansion is further shown to occur in the streamwise autocorrelation function in Fig. 3 (the locations of the two chosen points are highlighted in Fig. 2(a)).
Where the autocorrelation function is defined as: where γ = t i+1 − t i . From the streamwise autocorrelation function the sudden expansion is clearly highlighted by a jump in length of the integral time scales, The focus now concentrates on the first four planes, as this corresponds to the location of the sudden expansion of the mixing layer (as discussed in Section 1). These planes are highlighted in yellow in Fig. 1. A nomenclature for the 125 recirculation cells is also introduced, as shown in Fig. 4. The upstream recirculation zone is termed R1, the first downstream recirculation zone is termed R2 and the second downstream recirculation zone is termed R3. In Fig. 5 (a-c)   account for about 35% of the total variance of the flow. As a consequence it is     the temporal dimension can result in an overfit of the data. The spectra associated to each DMD calculation are shown in Fig. 9, where the frequencies relating to the extracted POD modes are highlighted in red. For the reader's reference, whilst in these spectra it is apparent that they are higher frequencies with high amplitudes, the majority of these modes relate to noise, originating from the 190 experimental data. Consequently highlighting that using a Fourier description based on a POD's temporal coefficients to select the spatial modes will mitigate against incorrectly choosing insignificant DMD modes with a high D α , created by experimental/background noise. As shown by Brevis and García-Villalba Chrisohoides et al. [9]). When (2) is sufficiently large enough it merges with (1) and advects along the mixing layer causing R3b to contract. This basic mechanism repeats itself periodically, leading to the observed dynamics, which have been documented for some time, but not previously explained satisfactorily. If the shallowness of the flow is decreased or the roughness is increased there will be an increase in the the size of eddies and the point of reattachment with the wall will change, which will affect the location of the sudden expansion. shown laboratory studies can be directly related to real flows [27]. However, the determination of the degree of roughness or shallowness leading the occurrence 260 and location of the sudden expansion of the mixing layer is beyond the scope of the present study.

Discussion
Whilst the occurrence of this phenomenon would seem to be problematic due to the increased momentum fluxes at the point of impingement, in fact there are a number of benefits associated with this mechanism. For example, in the 265 first downstream recirculation cell there are no large scale turbulent structures (see Fig. 13), i.e. a region in which momentum fluxes are minimal and a region not susceptible to erosion, sediment transport or mixing. This region therefore protects the near upstream region of the groyne and could offer a zone which could be beneficial for fish habitats and river restoration projects.

270
POD is well known in fluid mechanics and geophysics but relatively underutilised in hydraulics, DMD is a new technique with a great deal of work on it in the last 7 years, and that the fusion of the two in the way done in the present study is a further contribution. As shown in the present study these methods can easily extract the large scale turbulent dynamics which govern a 275 flow system. Whilst these methods may seem abstract, the processes which they are able to describe explain events which are of great significance in the mixing of pollutants, erosion and sediment transport. Furthermore, as shown in the present study, for complex systems with many intertwined turbulent processes, the Dynamic Mode Decomposition is able to describe these processes 280 individually, something which could be used to create simplistic models of highly complex systems. These combination of these methods is also of a benefit for noisy data. As shown by Wang et al. [54] low-order POD modes are unlikely to contain noise, therefore using low-order POD coefficients, as a search criterion, mitigates against the determination of noisy DMD modes. Although the present 285 study is based on a highly accurate PIV technique, as shown by Brevis and García-Villalba [6] this is not a requirement, and presented techniques could easily be applied data obtained from dye tracers / flow visualisations [11].

Conclusions
Using a reconstruction of modes from a Dynamic Mode Decomposition and