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: α=0.085. 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 mode). 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.