Galerkin-Laguerre Spectral Solution of Self-Similar Boundary Layer Problems

In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms

First Instability and Structural Sensitivity of the Flow Past Two Side-by-Side Cylinders

The onset of two-dimensional instabilities in the flow past two side-by-side circular cylinders is numerically investigated in the ranges 0.1 <= 6 <= 3 and Re < 100, with g being the non-dimensional gap spacing between the surfaces of the two cylinders and Re the Reynolds number. A comprehensive, global stability analysis of the symmetric base flow is carried out, indicating that three harmonic modes and one steady antisymmetric mode become unstable at different values of g and Re. These modes are known to promote distinct flow regimes at increasing values of g: single bluff-body, asymmetric, in-phase and antiphase synchronized vortex shedding. For each mode, the inherent structural sensitivity is examined in order to identify the core region of the related instability mechanism. In addition, by exploiting the structural sensitivity analysis to base flow modifications, a passive control strategy is proposed for the simultaneous suppression of the two synchronized shedding modes using two small secondary cylinders. Its effectiveness is then validated a posteriori by means of direct numerical simulations

Experimental assessment of drag reduction by traveling waves in a turbulent pipe flow

We experimentally assess the capabilities of an active, open-loop technique for drag reduction in turbulent wall flows recently introduced by Quadrio et al. [J. Fluid Mech., v.627, 161, (2009)]. The technique consists in generating streamwise-modulated waves of spanwise velocity at the wall, that travel in the streamwise direction. A proof-of-principle experiment has been devised to measure the reduction of turbulent friction in a pipe flow, in which the wall is subdivided into thin slabs that rotate independently in the azimuthal direction. Different speeds of nearby slabs provide, although in a discrete setting, the desired streamwise variation of transverse velocity. Our experiment confirms the available DNS results, and in particular demonstrates the possibility of achieving large reductions of friction in the turbulent regime. Reductions up to 33% are obtained for slowly forward-traveling waves; backward-traveling waves invariably yield drag reduction, whereas a substantial drop of drag reduction occurs for waves traveling forward with a phase speed comparable to the convection speed of near-wall turbulent structures. A Fourier analysis is employed to show that the first harmonics introduced by the discrete spatial waveform that approximates the sinusoidal wave are responsible for significant effects that are indeed observed in the experimental measurements. Practical issues related to the physical implementation of this control scheme and its energetic efficiency are briefly discussed.Comment: Article accepted by Phys. Fluids. After it is published, it will be found at http://pof.aip.or

Centre-Manifold Reduction of Bifurcating Flows

In this paper we describe a general and systematic approach to the centre-manifold reduction and normal form computation of flows undergoing complicated bifurcations. The proposed algorithm is based on the theoretical work of Coullet & Spiegel (SIAM J. Appl. Maths, vol. 43(4), 1983, pp. 776821) and can be used to approximate centre manifolds of arbitrary dimension for large-scale dynamical systems depending on a scalar parameter. Compared with the classical multiple-scale technique frequently employed in hydrodynamic stability, the proposed method can be coded in a rather general way without any need to resort to the introduction and tuning of additional time scales. The method is applied to the dynamical system described by the incompressible NavierStokes equations showing that high-order, weakly nonlinear models of bifurcating flows can be derived automatically, even for multiple codimension bifurcations. We first validate the method on the primary Hopf bifurcation of the flow past a circular cylinderand after we illustrate its application to a codimension-two bifurcation arising in the flow past two side-by-side circular cylinder

On the Origin of the Flip-Flop Instability of two Side-by-Side Cylinder Wakes

In this work the flip-flop instability occurring in the flow past two side-by-side circular cylinders is numerically investigated within the range of non-dimensional gap spacing 0.6<g<1.4 and Reynolds number 50<Re\leq 90. The inherent two-dimensional flow pattern is characterized by an asymmetric unsteady wake (with respect to the horizontal axis of symmetry) with the gap flow being deflected alternatively toward one of the cylinders. Such behaviour has been ascribed by other authors to a bistability of the flow, and therefore termed flip-flop. In contrast, the simulations performed herein provide new evidence that at low Reynolds numbers the flip-flopping state develops through an instability of the in-phase synchronized vortex shedding between the two cylinder wakes. This new scenario is confirmed and explained by means of a linear global stability investigation of the in-phase periodic base flow. The Floquet analysis reveals indeed that a pair of complex-conjugate multipliers becomes unstable having the same low frequency as the gap flow flip-over. The neutral curve of this secondary instability is tracked within the above range of gap spacing. The spatiotemporal shape of the unstable Floquet mode is then analysed and its structural sensitivity is considered in order to identify the 'core' region of the flip-flop instability mechanism

An Almost Subharmonic Instability in the Flow Past Rectangular Cylinders

The three-dimensional instability of the flow past a 5 :1 rectangular cylinder is investigated via Floquet analysis and direct numerical simulations. A quasi-subharmonic (QS) unstable mode is detected, marking an important difference with the flow past bodies with lower aspect ratio and/or with smooth leading edge. The QS mode becomes unstable at Reynolds number (based on the cylinder thickness and free-stream velocity) Re approximate to 480; its spanwise wavelength is approximately three times the cylinder thickness. The structural sensitivity locates the wavemaker region over the longitudinal sides of the cylinder, indicating that the instability is triggered by the mutual inviscid interaction of vortices generated by the leading edge shear layer