Investigation On The 3D Transition Features In The Wake Of A Cylinder

The flow past a uniform circular cylinder is studied in detail by direct numerical simulations of three-dimensional incompressible Navier-Stokes equations. The features and dynamics for various Reynolds numbers in the three-dimensional transition regime of the cylinder wake are investigated. The 3rd-order splitting algorithm based on the mixed stiffly stable scheme is employed in the temporal discretization of the N-S equations and the mixed Fourier-spectral-spectral-element method in the spatial discretization.The errors in calculating derivatives for the GLL collocation points are evaluated, which can be alleviated from O((N4) to O((N2) by the double-precision method proposed in the present paper, where ( denotes the machine precision and N the order of the interpolation polynomials in the elements. The exact solutions of the N-S equations are employed to verify the algorithm and the program. The two-dimensional flows past both a still cylinder and a rotating cylinder are computed. Our numerical results show good conformance with that from the previous experimental and numerical studies. After the verification, the three-dimensional numerical simulations of the cylinder wake for Re = 200, 250 and 300 are performed.Re = 200 is just beyond the critical Reynolds number of mode A. The numerical results of the present paper indicate that the near wake at this supercritical Reynolds number is in three-dimensional quasi-periodic laminar state with transitional behaviors. The spanwise characteristic length determines the transition features and global properties of the wake. Especially for the specific spanwise characteristic length linear stable mode can dominate the wake in place of mode A and determine the spanwise phase difference of the primary vortices shedding.At Re = 250, mode B is subcritical. The present studies suggest that mode A spontaneously emerges in the wake preceding all the other spanwise modes. Then it excites the linear stable mode B through the nonlinear interactions among the various spanwise modes. Eventually mode A and B coexist in the wake, which confirms the previous studies. Besides, the present paper finds that downstream the streamwisevortices evolve into a new type of mode - "dual vortex pair mode". An independent low frequency fm, which result in the irregularity of the temporal signals, other than the vortex shedding frequency is also identified.When Re = 300, mode A and B are both unstable. Due to its higher growth rate mode B replaces the 2-D wake first. Then with the growth of mode A, the two modes coexist in the wake. From the surface of the cylinder to the formation region of the vortices then to the wake downstream, the structure of the streamwise vortices change from mode A to mode B, dual vortex pair mode then mode A again.Preliminary studies are performed on the effect of the non-uniform stream with cosine form velocity profile in spanwise direction on the three-dimensional transition of the cylinder wake. The parameters taken into consideration include the Reynolds number, the wavelength and the amplitude of the stream profile. The numerical results indicate that in a quite long period of evolution the wavelength of the stream profile determines the flow. In the wake the spanwise modes with the identical wavelength and the harmonic waves are excited, while the other modes are strongly suppressed. The similarity between the streamwise vortex structure in the wake and mode A for the uniform stream supports the opinion that mode A derives from the three-dimensionality in the bypassing process of the flow around the cylinder.