Scale-by-scale dynamic behaviour of a passive scalar affected by coherent motion in a cylinder wake

We examine the dynamic interactions between the large-scale coherent motion and the small-scale turbulence in the passive scalar field of a circular cylinder wake, where the coherent motion exhibits strong periodicity. A combination of four Xwires and four cold wires was used to simultaneously measure the three velocity and temperature fluctuations at nominally the same location. Measurements were taken at x/d = 10, 20 and 40 in the mean shear plane at Reynolds number 2500, based on the cylinder diameter d and the free-stream velocity. The phase-averaging technique is used to distinguish the large-scale coherent motion from the stochastic motion, enabling the construction of phase-averaged structure functions of the passive scalar in the scale phase plane. The maximum of the coherent scalar θ˜ closely aligns with the minima of the phase-averaged strain S and the vortex centre, suggesting that heat is contained within the interior of the vortex. The scale-by-scale distributions of the scalar variance and the streamwise velocity variance exhibit a similar phase dependence associated with the coherent motion. This dependence is perceptible even at the smallest scales. However, as the distance from the cylinder increases, the perceivable scale range decreases and eventually disappears. An expression is formulated to describe the time-averaged second-order structure function of coherent scalar and the time-averaged second-order mixed structure function between the coherent scalar and coherent streamwise velocity at x/d = 10 and 20, where the coherent motion is prominent. Furthermore, the scale-by-scale contribution of the coherent scalar variance to the total scalar variance is evaluated. Also, we derive the scale-by-scale scalar variance transport equations that account for the coherent motion in both general and isotropic formulations. Assuming local isotropy, it is found that the equation agrees approximately with the experimental data across all scales at x/d = 40. Finally, the differences between the scaleby-scale transport equation for the stochastic scalar variance and that for the stochastic turbulent kinetic energy are discussed.