Numerical investigation of transitional and weak turbulent flow past a sphere

Abstract

This work reports results of numerical simulations of viscous incompressible flow past a sphere. The primary objective is to identify transitions that occur with increasing Reynolds number, as well as their underlying physical mechanisms. The numerical method used is a mixed spectral element/Fourier spectral method developed for applications involving both Cartesian and cylindrical coordinates. In cylindrical coordinates, a formulation, based on special Jacobi-type polynomials, is used close to the axis of symmetry for the efficient treatment of the ‘pole’ problem. Spectral convergence and accuracy of the numerical formulation are verified. Many of the computations reported here were performed on parallel computers. It was found that the first transition of the flow past a sphere is a linear one and leads to a three-dimensional steady flow field with planar symmetry, i.e. it is of the ‘exchange of stability’ type, consistent with experimental observations on falling spheres and linear stability analysis results. The second transition leads to a single-frequency periodic flow with vortex shedding, which maintains the planar symmetry observed at lower Reynolds number. As the Reynolds number increases further, the planar symmetry is lost and the flow reaches a chaotic state. Small scales are first introduced in the flow by Kelvin–Helmholtz instability of the separating cylindrical shear layer; this shear layer instability is present even after the wake is rendered turbulent.

Low-dimensional dynamics of a turbulent axisymmetric wake

“…Similar regimes are observed in the wake behind a sphere (Pier 2008;Tomboulides & Orszag 2000) and a disk (Natarajan & Acrivos 1993) suggesting that the transitional behaviour of axisymmetric wakes follows a similar scenario during the initial stages at low Reynolds numbers of O(100).…”

Low-frequency unsteadiness in the vortex formation region of a circular cylinder

“…Berger et al 18 studied the formation of coherent structures in the wake of a disk and a sphere at Reynolds numbers between 1.5 × 10 4 < Re < 3 × 10 5 , and found what they called a pumping mechanism of the recirculation bubble with a very low-frequency. In the case of the sphere, the existence of such low-frequency modulation was observed in the energy spectra from numerical experiments of Tomboulides and Orszag, 19 but recently has been also measured by means of DNS at Re = 3700 by Rodríguez et al 20 Najjar and Balachandar 21 also observed a low-frequency unsteadiness in the wake of a normal flat plate, finding a high-drag regime with high coherence in the span-wise vortices and a low-mean drag regime where less coherent vortices were formed. Later, Miau et al 22 found such variations in the wake of a trapezoidal cylinder and a circular cylinder at Reynolds numbers above 10 4 and showed that they were associated with the unsteady variations of the vortex formation length.…”

Vortex-induced vibration of a neutrally buoyant tethered sphere

“…The wake transitions for another widely studied bluff body, the sphere, are remarkably different to those of the circular cylinder [6,12,19,18]. A major difference in the wake transition behaviour of the sphere and the circular cylinder wake is that the sphere wake becomes asymmetrical prior to a transition to unsteady flow, whereas the cylinder wake does not become asymmetrical until the wake goes unsteady [20].…”

“…A major difference in the wake transition behaviour of the sphere and the circular cylinder wake is that the sphere wake becomes asymmetrical prior to a transition to unsteady flow, whereas the cylinder wake does not become asymmetrical until the wake goes unsteady [20]. For the sphere wake, the transition from attached to separated flow at the rear of the sphere has been found from direct numerical simulations to be Re 1 = 20 [19,6]. As the Reynolds number increased, the wake remains steady and axisymmetric up to Re 2 = 211 [6].…”