Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flow modifications

The question of optimal spanwise-periodic modification for the stabilisation of spanwise-invariant flows is addressed. A 2nd-order sensitivity analysis is conducted for the linear temporal stability of parallel flows U0 subject to small-amplitude spanwise-periodic modification e*U1, e<<1. Spanwise-periodic modifications have a quadratic effect on stability, i.e. the 1st-order eigenvalue variation is zero. A 2nd-order sensitivity operator is computed from a 1D calculation, allowing one to predict how eigenvalues are affected by any U1, without actually solving for modified eigenvalues/eigenmodes. Comparisons with full 2D stability calculations in a plane channel flow and in a mixing layer show excellent agreement. Next, optimisation is performed on the 2nd-order sensitivity operator: for each eigenmode streamwise wavenumber and base flow modification spanwise wavenumber b, the most stabilising profiles U1 are computed, together with lower bounds for the variation in leading eigenvalue. These bounds increase like b^-2 as b goes to 0, yielding a large stabilising potential. However, 3D modes with wavenumbers |b0|=b and b/2 are destabilised, thus larger control wavenumbers should be preferred. The modification U1 optimised for the most unstable streamwise wavenumber has a stabilising effect on other streamwise wavenumbers too. Finally, the potential of transient growth to amplify perturbations and stabilise the flow is assessed. Combined optimal perturbations that achieve the best balance between transient linear amplification and flow stabilisation are determined. In the mixing layer with b<1.5, these combined optimal perturbations appear similar to transient growth-only optimal perturbations, and achieve a more efficient overall stabilisation than optimal 1D and 2D modifications computed for stabilisation only. This is consistent with the efficiency of streak-based control strategies.Comment: 23 pages, 15 figure

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared to the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinise the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points and it pinpoints the crucial need of considering this effect when investigating critical transitions

Stability and dynamics of the laminar flow past rectangular prisms

The laminar flow past rectangular prisms is studied in the space of length-to-height ratio ($1 \le L/H \le 5$), width-to-height ratio ($1.2 \le W/H \le 5$) and Reynolds number ($Re \lessapprox 700$). The primary bifurcation is investigated with linear stability analysis. For large $W/L$ it consists of an oscillating mode breaking the top/bottom planar symmetry. For smaller $W/L$ the flow becomes first unstable to stationary perturbations, and the wake experiences a static deflection, vertical for intermediate $W/L$ and horizontal for small $W/L$. Weakly nonlinear analysis and nonlinear direct numerical simulations are used for $L/H = 5$ and larger $Re$. For $W/H = 1.2$ and $2.25$, after the primary bifurcation the flow recovers the top/bottom planar symmetry but loses the left/right one, via supercritical and subcritical pitchfork bifurcations, respectively. Further increasing $Re$, the flow becomes unsteady and oscillates around either the deflected (small $W/H$) or the non-deflected (intermediate $W/H$) wake. For intermediate $W/H$ and $R$e, a periodic and fully symmetric regime is detected, with hairpin vortices shed from the top and bottom leading-edge (LE) shear layers; its triggering mechanism is discussed. At large $Re$ and for all $W/H$, the flow approaches a chaotic state characterised by the superposition of different modes: shedding of hairpin vortices from the LE shear layers, and wake flapping in the horizontal and vertical directions. In some portions of the parameter space the different modes synchronise, giving rise to periodic regimes also at relatively large $Re$.Comment: 55 pages, 37 figure

Controlled reattachment in separated flows: a variational approach to recirculation length reduction

A variational technique is used to derive analytical expressions for the sensitivity of recirculation length to steady forcing in separated flows. Linear sensitivity analysis is applied to the two-dimensional steady flow past a circular cylinder for Reynolds numbers $40 \leq Re \leq 120$, both in the subcritical and supercritical regimes. Regions which are the most sensitive to volume forcing and wall blowing/suction are identified. Control configurations which reduce the recirculation length are designed based on the sensitivity information, in particular small cylinders used as control devices in the wake of the main cylinder, and fluid suction at the cylinder wall. Validation against full non-linear Navier-Stokes calculations shows excellent agreement for small-amplitude control. The linear stability properties of the controlled flow are systematically investigated. At moderate Reynolds numbers, we observe that regions where control reduces the recirculation length correspond to regions where it has a stabilising effect on the most unstable global mode associated to vortex shedding, while this property does not hold any more at larger Reynolds numbers.Comment: 17 pages, 11 figure

From thin plates to Ahmed bodies: linear and weakly non-linear stability of rectangular prisms

We study the stability of laminar wakes past three-dimensional rectangular prisms. The width-to-height ratio is set to $W/H=1.2$, while the length-to-height ratio $1/6<L/H<3$ covers a wide range of geometries from thin plates to elongated Ahmed bodies. First, global linear stability analysis yields a series of pitchfork and Hopf bifurcations: (i) at lower Reynolds numbers $Re$, two stationary modes, $A$ and $B$, become unstable, breaking the top/bottom and left/right planar symmetries, respectively; (ii) at larger $Re$, two oscillatory modes become unstable and, again, each mode breaks one of the two symmetries. The critical $Re$ of these four modes increase with $L/H$, qualitatively reproducing the trend of stationary and oscillatory bifurcations in axisymmetric wakes (e.g. thin disk, sphere and bullet-shaped bodies). Next, a weakly non-linear analysis based on the two stationary modes $A$ and $B$ yields coupled amplitude equations. For Ahmed bodies, as $Re$ increases state $(A,0)$ appears first, followed by state $(0,B)$. While there is a range of bistability of those two states, only $(0,B)$ remains stable at larger $Re$, similar to the static wake deflection (across the larger base dimension) observed in the turbulent regime. The bifurcation sequence, including bistability and hysteresis, is validated with fully non-linear direct numerical simulations, and is shown to be robust to variations in $W$ and $L$ in the range of common Ahmed bodies