Dynamical interactions between the coherent motion and small scales in a cylinder wake

Abstract

AbstractMost turbulent flows are characterized by coherent motion (CM), whose dynamics reflect the initial and boundary conditions of the flow and are more predictable than that of the random motion (RM). The major question we address here is the dynamical interaction between the CM and the RM, at a given scale, in a flow where the CM exhibits a strong periodicity and can therefore be readily distinguished from the RM. The question is relevant at any Reynolds number, but is of capital importance at finite Reynolds numbers, for which a clear separation between the largest and the smallest scales may not exist. Both analytical and experimental tools are used to address this issue. First, phase-averaged structure functions are defined and further used to condition the RM kinetic energy at a scale $r$ on the phase $\phi $ of the CM. This tool allows the dependence of the RM to be followed as a function of the CM dynamics. Scale-by-scale energy budget equations are established on the basis of phase-averaged structure functions. They reveal that energy transfer at a scale $r$ is sensitive to an additional forcing mechanism due to the CM. Second, these concepts are tested using hot-wire measurements in a cylinder wake, in which the CM is characterized by a well-defined periodicity. Because the interaction between large and small scales is most likely enhanced at moderate/low Reynolds numbers, and is also likely to depend on the amplitude of the CM, we choose to test our findings against experimental data at $R_{\lambda } \sim 10^2$ and for downstream distances in the range $10 \leq x/D \leq 40$. The effects of an increasing Reynolds number are also discussed. It is shown that: (i) a simple analytical expression describes the second-order structure functions of the purely CM. The energy of the CM is not associated with any single scale; instead, its energy is distributed over a range of scales. (ii) Close to the obstacle, the influence of the CM is perceptible even at the smallest scales, the energy of which is enhanced when the coherent strain is maximum. Further downstream from the cylinder, the CM clearly affects the largest scales, but the smallest scales are not likely to depend explicitly on the CM. (iii) The isotropic formulation of the RM energy budget compares favourably with experimental results.