Longitudinal vortex structures in a cylinder wake

Recently, flow visualizations by Wei and Smith,’ Williamson,’ Welsh et a1.,3 and Bays-Muchmore and Ahmed4 have shown that three-dimensional (3-D) vertical structures develop in the wake of a bluff body. The 3-D vertical structures were found to include pairs of counterrotating longitudinal vortices superimposed on the nominally two-dimensional (2-D) Kirmin vortex street as speculated by Grant’ decades ago. The vortices are similar to the 3-D flow structures observed by Bernal and Roshko” in plane mixing layers. While the existence and the topological details of the longitudinal vortex structures have been recognized through visual observation, the physical characteristics and dynamical properties of the structures remain to be explored through quantitative measurement. In the present study, digital particle image velocimetry [PIV) has been used to measure the instantaneous velocity field in a transverse plane (the x-z plane, where the x axis is in the flow direction and the z axis is parallel to the cylinder axis) behind the circular cylinder. The principle of PIV is simple: the flow field, seeded with particles, is illuminated by a pulsed laser light sheet, the displacements of the double or multiply exposed particles are recorded and then analyzed using specially developed software to obtain instantaneous velocity distributions. PIV techniques have attracted substantial attention in recent years and comprehensive reviews have been provided by Adrian7 and Buchhave.s In this application, single frame and multiply exposed digital particle images were acquired using a CCD camera with a spatial resolution of 1280X1024 pixel. The experiment was carried out in a low turbulence water tunnel at a Reynolds number of 52.5 based on cylinder diameter. This Reynolds number was chosen to be well above the transitional range noted by Williamson’ for the onset of the longitudinal vortices, so that the structures were expected to be fully developed and representative of a wider range of Reynolds number. The measurements were conducted in a transverse plane 2D behind the back of the cylinder and covered an area of approximately 2.8X2.20. The in-plane velocity vectors, with approximately 2000 points per frame, were obtained using Young’s fringe patterns, calculated from 2-D FFT over small interrogation windows overlapped at 50%. The overall velocity measurement uncertainty is estimated to be 4% and the vorticity error 15%-20%, at the 95% confidence level. A typical hydrogen-bubble flow visualization pattern is presented in Fig. l(a). It can be seen that mushroom-type structures, indicated by the arrow, develop in the near wake behind a circular cylinder, implying the existence of the counter-rotating longitudinal vortices in the cylinder wake. An instantaneous velocity distribution sampled at an arbitrary instant is presented in Figs. l(b) and l(c), with the frame of reference moving at a speed approximately equal to the eddy convection velocity lJ,=60% U,,, where U, is the free-stream velocity. The cross-sectional streamline patterns were obtained by integrating the velocity field. The spiraling of the streamline patterns near vortex centers is indicative of flow three-dimensionalities, i.e., flow motions out of the measurement plane. From a survey of over 130 vortex pairs contained in 50 instantaneous velocity data frames, it was found that the streamlines usually spiral in around vortex centers. This suggests that the vortices are experiencing expansive strain fields perpendicular to the measurement plane. Transverse vorticity has heen calculated from the discrete velocity data. Figure 2 shows fiuctuations of the maximum and minimum transverse vorticity component of vortex pairs as the test is repeated, where free-stream velocity and cylinder diameter are used for normalization: $=oJ( UOID), where ~0~ is the transverse vorticity component. The mean of both positive and negative vorticity is approximately equal, &=+7.3. Measurements made using the same PIV technique show that the maximum spanwise vorticity of a shed Strouhal vortex is approximately Q =4-5 at Re=525. Therefore the maximum vorticity of the longitudinal vortices is greater than that of the spanwise vortices. This result is perhaps expected, as longitudinal vortices are thought to be stretched by spanwise vortices, as suggested by Wei and Smith‘ and Meiburg and Lasheras,” among others. If the longitudinal vortices are hypothesized to originate from the spanwise vortices, then a higher vorticity of the longitudinal vortices is consistent with the vortex stretching theory. The circulation of a longitudinal vortex has been estimated by integrating vorticity over the area in which & is greater than 10% of ty max. The mean circulation of longitudinal vortices was found to be r/( U,D) =0.39 with a standard deviation of 0.14, and the mean circulation of K&m& vortices was found to be r,,/( lJ,Dj=3.5 with a standard