On the onset of secondary stability in the wake of two rotating cylinders in tandem arrangement

Abstract

Fluid dynamic transport around a body naturally observe unstable flow behind the body including different patterns of vortical shedding. But simultaneous imposition of rotation to vortex shedding bluff body results in flow stabilization. Eventually, steady flow behind a bluff body is obtained by precisely controlling its overall rotation. In this study critical rotational speeds for two circular cylinders in tandem arrangement are focused for the initiation and termination of secondary vortices in an unconfined flow. Also, the effect of altering the cylinders gap ratios and rotation rates over the obtained flow regimes and respective force coefficients is studied in detail. Two circular cylinders are subjected to a uniform free stream flow of 100 Reynold number (ρ𝑈∞D/µ) with varying gap ratios of 1.5, 2 and 4 L/D (center to center) each. At each angular rotation (ωD/𝑈∞) two modes of rotation are observed, firstly when both the cylinders are made to rotate in the same direction (anticlockwise rotation in this study), secondly when upstream cylinder is rotated in anticlockwise direction and downstream cylinder is rotated in clockwise direction, thus capturing all the possible combinations of two tandem rotating cylinders. Non-dimensional angular rotations (α) applied to cylinders vary all the way from stationary to the specific α where the secondary instability subsides. Multiple flow regimes along with their sub-divisions are outlined for co-rotation and counter-rotation of two circular cylinders in tandem arrangement depending on the vortical shed pattern. For co-rotation transitions in flow regimes are noticed from 0 to 6α. Gap ratios of 1.5 L/D and 2 L/D show Solitary Periodic (SP) flow where vortices shed in a periodic manner at stationary and low α values. But for a higher gap ratio of 4 L/D, alternate co-shedding (AC) flow is noticed at stationary and low α values where both the cylinders show distinct vortical shedding. Increasing α transits the flow into steady flow referred as SS-Ⅰ flow regime, where shear layers shed from the combined system of cylinders in a constant manner. Further increment in α results in the transition of flow from SS-Ⅰ to secondary unstable state i.e. single rotating bluff body (SRB) flow. SRB flow is sub-divided into two categories termed as integrated and segregated SRB flow obtained at low and high gap ratios respectively. Finally supplementary α causes wrapping of shear layers around the cylinder with overall steady behavior and thus this flow regime is denoted by SS-Ⅱ flow regime. It is observed that with thex increase of gap ratio the secondary vortices show delayed transition between distinct flow regimes along with delayed starting and ending of secondary vortices. For counter rotation of tandem circular cylinders different gap ratios demonstrate distinct flow regimes. Gap ratio of 1.5 L/D shows only SP and SS-Ⅰ flow regimes. 2 L/D gap ratio illustrates development of three distinct sub-divisions of fundamental SP, SS-Ⅰ, SRB, and SS-Ⅱ flow regimes represented as steady state Ⅰ & Ⅱ (SS-Ⅰ & Ⅱ), secondary unstable - inverted rotation (SU-IR), and secondary unstable (SU-SS-Ⅱ). Where the upstream and downstream cylinders show dissimilar shedding of vortices or shear layers. Moreover, the transition of these flow regimes from one to another is noticed to be at immensely higher α values. 4 L/D gap ratio demonstrates generic development of AC, SS-Ⅰ, SRB and SS-Ⅱ flow regimes. Except for the 2 L/D case, simulations of various gap ratios for counter-rotating cylinders demonstrate that the α range extends up to 6.25α for the complete development of all flow states. Flow transitions at atypically higher values of 12α are obtained for gap ratio of 2 L/D. Force coefficients for both co-rotation and counter-rotation of cylinder show an increase in magnitude with increase of applied α. Co-rotating and counter-rotating, tandem cylinders show repelling and attractive nature towards each other owing to the rotation derived presence and absence of stagnation point between both the cylinders respectively. Presence of vortex shedding pattern in the force plots is traced using standard deviation of force coefficient (𝜎𝐶𝐿 and 𝜎𝐶𝐷) plots with reference to the mean force coefficient values. In comparison to the co-rotating cylinder, counter-rotation shows predominant inclination towards stable flow behavior.