A linear approach for the evolution of coherent structures in shallow mixing layers

Abstract

The development of large coherent structures in a shallow mixing layer is analyzed. The results are validated with experimental data obtained from particle tracking velocimetry. The mean flow field is modeled using the self-similarity of the velocity profiles. The characteristic features of the down-stream development of a shallow mixing layer flow, like the decrease of the velocity difference over the mixing layer, the decreasing growth of the mixing layer width, and the transverse shift of the center of the mixing layer layer are fairly well represented. It turned out that the entrainment coefficient could be taken constant, equal to a value obtained for unbounded mixing layers: ␣ϭ0.085. Linearization of the shallow water equations leads to a modified Orr-Sommerfeld equation, with turbulence viscosity and bottom friction as dissipative terms. Growth rates are obtained for each position downstream, using the model for the mean flow field. For a given energy density spectrum at the inflow boundary, integration of the growth rates along the downstream direction yields the spectra at various downstream positions. These spectra provide a measure for the intensity and the length scale of the coherent structures ͑the dominant mode͒. The length scales found are in good agreement with the measured ones. The length scale of the most unstable mode appears much larger than the length scale of the dominant mode. Obviously, the longevity of the coherent structures plays a significant role. Three growth regimes can be distinguished: in the first regime the dominant mode is growing, in the second regime the dominant mode is dissipating, but other modes are still growing, and in the third regime all modes are dissipating. It is concluded that the development of the coherent structures in a shallow mixing layer can fairly well be described and interpreted by the proposed linear analysis.

Large scale PIV-measurements at the surface of shallow water flows

“…Shallow flow phenomena can be found in rivers, coastal zones or the atmosphere, where the dominating processes are two-dimensional [1]. Large coherent motions, that are inititated, for instance, in shallow wake flows behind obstacles [2] or in shallow mixing layers [3][4][5] strongly influence the dynamics of such a system and are, therefore, a determining factor for the mixing and transport processes. Measurement of the velocity field describing the generation and the evolution of these coherent motions is the major aim of this investigation.…”

Stability of shallow mixing layers with asymmetric base flow profile

“…It should be also pointed out that the base flow used in the paper is obtained from averaged long-term evolution of the flow, so that it includes nonlinear effects and changes in the initial structure of the mixing layer. However, such an approach is used earlier in the literature [8] and found to be useful in characterizing the development of coherent structures (at least at the initial stage). More detailed parametric study is needed in order to fully assess the effect of asymmetry of the base flow on the stability characteristics of shallow mixing layers.…”

The Relevance of a Back-Scatter Model for Depth-Averaged Flow Simulation

“…[34]. The arrows indicate the mean streamwise velocity 1.2 Linear behaviour of unstable flows in a shallow mixing layer A study by [33] has revealed that the spectral distribution of the large scale motions of a shallow mixing layer can be predicted from a linear stability analysis. First, the mean flow was predicted, based on the self-similarity method proposed by [5].…”

“…This implies also that the intensity level from which the Values measured at the free-surface (dots) are compared with a prediction derived from a linear stability analysis (lines). The straight horizontal line represents the uniform energy level imposed at the inflow [33] structures start to grow are of key importance for their strength. The turbulence properties of the ambient flow therefore play an important role in the energy transfer between the turbulent scales in the mixing layer and should thus be accounted for.…”

“…This can be considered as the non-linear implementation of the above described linear stability analysis [33]. Second, a backscatter model for the background quasi 2D turbulence will be described that includes the characteristics of large turbulence structures as they appear in a uniform open channel flow.…”