Numerical Study On The Near-Wake Behind A Circular Disk

The flow over a circular disk has been the subject of many numerical and experimental studies, due to its relevance in many aero- and hydrodynamic applications, e.g. parachute and flame holder. Meanwhile, the wakebehind a circular disk is a typical flow of axisymmetric bodies with sharp edges, and it informs our understanding of the shear-layer separation, transition to turbulence, and wake development in the wake of three-dimensional bluff bodies. However, the wakes behind circular disks are much less studied compared with other basic geometries, e.g. spheres and cylinders. In the present work, a numercial investigation on the near wake of a circular disk with aspect ratio of χ=5 is performed. The main work and the correspoinding results are as follows:Firstly, the wake evolutions behind a circular disk at low Reynolds numbers (Reynolds number of 115 to 103) are numerically studied. When Re<152, the wake stays steady and no vortex shedding occurs. A regular bifurcation leading to two-thread wake occurs at Re=120. When Re increases to 152, the hairpin vortices begin to be periodically shed in a regular pattern. A reflectional-symmetry-breaking (RSB) mode is observed for Re from 152 to 155, while an unsteady planar-symmetry mode is captured for Re from 156 to 265. It is found that the hairpin vortices are always shedding in a fixed orientation for RSB mode, while shedding from diametrically opposite orientations at unsteady planar-symmetry mode. When Re increases to 265, the hairpin vortices are periodically shed in an irregular pattern and a new ’4L’ wake mode is identified. When Re increases to 650, the shear-layer instability starts to be detected in the shear layer. The critical Reynolds number of the shear-layer instability in the circular disk wake is firstly identified in the range of 601～650. The cylindrical shear layer surrounding the vortex formation zone rolls up to form oblate ring vortices. The hairpin vortices break into pieces and small-scale vortices associated with the shear-layer instability are observed.Secondly, a numerical study on the turbulent wake behind a circular disk at Re=104 is performed. A large waviness of the wake coherent structures is observed and further studied with the definition of the wake positions. It is suggested that the coexistence of lots of irregular helical/helical-like structures in the wake leads to the waviness. The traces of the wake positions in phase diagram could give a good description of the irregular motion of the large-scale vortex shedding location, which occurs at the same large-scale vortex shedding frequency. Through two different conditional averagings, it is found that there exists two representative coherent structures in the turbulent disk wake:one is planar-symmetric coherent structure, which bears a remarkable resemblance to the wake structure at unsteady planar-symmetry mode for low Reynolds numbers; while the other one is unsymmetric coherent structure, which bears a remarkable resemblance to the wake structure at RSB mode.Thirdly, the possible physical links between the turbulent disk wake and the laminar/transitional wakes are numerically explored using fully 3D proper orthogonal decomposition (POD). The Reynolds numbers considered in this study (Re=152,170, 300 and 3×103) cover the laminar/transitional states, i.e., RSB mode, unsteady planar-symmetry mode and weakly chaotic state, and a turbulent wake. Through analysis of the spatial POD modes at different wake states, it is found that a ’planar-symmetric vortex shedding’mode characterized by the first mode pair is persistent at all the states. When the wake develops into a weakly chaotic state, a new vortex shedding mode characterized by the second mode pair begins to appear and completely forms at the turbulent wake of Re=3×103, i.e.,’planar-symmetry-breaking vortex shedding’mode. On the other hand, the coherent structure at the turbulent wake extracted from the first two POD modes shows a good resemblance to the wake configuration at unsteady planar-symmetry mode, while the coherent structure reconstructed from the first four POD modes shows a good resemblance to the wake configuration at RSB mode. The present results indicate that the dynamics or flow instabilities (e.g. vortex shedding regimes) observed at laminar/transitional RSB and unsteady planar-symmetry modes are still preserved at a turbulent disk wake.Lastly, the low-frequency characteristics in the wake behind a circular disk are investigated. The Reynolds numbers considered in this study are Re=250,300,3×103 and 104. The Fourier spectra analysis of the velocity at different positions, suggests that together with the natural vortex-shedding frequency and the shear-layer Kelvin-Helmholtz instability frequency, several much lower frequencies are also detected when the Reynolds number increases to 300, namely Stp≈0.03, Str≈0.02 and its second harmonic at about 0.04 and a low frequency of about 0.05. Considering the limited time length of the numerical data, time-frequency analysis based on wavelet transform is then performed. It is found that the local frequencies of the amplitude peaks for the large-scale vortex shedding and shear-layer Kelvin-Helmholtz instability vary irregularly with time, indicating that the traditional Fourier analysis of the numerical signals would not be representative. Nevertheless, the low frequencies Stp≈0.03 and Str≈0.02 are clearly confirmed. The frequencies of about 0.04 and 0.05 are observed to occur in a few time periods but with much weaker magnitude. It is found that the low-frequency signals keep nearly invariant as Reynolds number increases. The physical origins of the low frequencies Stp≈0.03 and Str≈0.02 are then investigated. The physical origin of Stp≈0.03 is due to a low-frequency pumping motion of the recirculation bubble, and the Str≈0.02 is due to a low-frequency modulation in the rotation of the azimuthal location of large-scale vortex shedding (i.e., the changing of the rotation direction). To characterize the low-frequency unsteadiness of the recirculation bubble, the temporal evolution of the low-pass filtered flow field is presented. It is observed that the pumping motion of the recirculation bubble is associated with the growing and shedding of the large toroidal vortices formed in the bubble, which is closely related to the behavior of the separated shear layer. The separated shear layer rolls up later in the state with longer recirculation bubble, whereas it rolls up earlier in the state with shorter bubble.