14 research outputs found
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
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 , while the
length-to-height ratio 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 , two stationary modes, and , become unstable, breaking the
top/bottom and left/right planar symmetries, respectively; (ii) at larger ,
two oscillatory modes become unstable and, again, each mode breaks one of the
two symmetries. The critical of these four modes increase with ,
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 and
yields coupled amplitude equations. For Ahmed bodies, as increases state
appears first, followed by state . While there is a range of
bistability of those two states, only remains stable at larger ,
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 and in the
range of common Ahmed bodies
Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flowmodifications
The question of optimal spanwise-periodic modification for the stabilisation of spanwise-invariant flows is addressed. A second-order sensitivity analysis is conducted for the linear temporal stability of parallel flows subject to small-amplitude spanwise-periodic modification .It is known that spanwise-periodic flow modifications have a quadratic effect on stability properties, i.e. the first-order eigenvalue variation is zero, hence the need for a second-order analysis. A second-order sensitivity operator is computed from a one-dimensional calculation, which allows one to predict how eigenvalues are affected by any flow modification , without actually solving for modified eigenvalues and eigenmodes. Comparisons with full two-dimensional stability calculations in a plane channel flow and in a mixing layer show excellent agreement. Next, optimisation is performed on the second-order sensitivity operator: for each eigenmode streamwise wavenumber and base flow modification spanwise wavenumber , the most stabilising/destabilising profiles are computed, together with lower/upper bounds for the variation in leading eigenvalue. These bounds increase like as goes to zero, thus yielding a large stabilising potential. However, three-dimensional modes with wavenumbers , are destabilised, and therefore larger control wavenumbers should be preferred. The most stabilising optimised for the most unstable streamwise wavenumber has a stabilising effect on modes with other values too. Finally, the potential of transient growth to amplify perturbations and stabilise the flow is assessed with a combined optimisation. Assuming a separation of time scales between the fast unstable mode and the slow transient evolution of the optimal perturbations, combined optimal perturbations that achieve the best balance between transient linear amplification and stabilisation of the nominal shear flow are determined. In the mixing layer with , these combined optimal perturbations appear similar to transient growth-only optimal perturbations, and achieve a more efficient overall stabilisation than optimal spanwise-periodic and spanwise-invariant modifications computed for stabilisation only. These results are consistent with the efficiency of streak-based control strategie
Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flow modifications
Homogenization-based design of microstructured membranes: Wake flows past permeable shells
A formal framework to characterize and control/optimize the flow past permeable membranes by means of a homogenization approach is proposed and applied to the wake flow past a permeable cylindrical shell. From a macroscopic viewpoint, a Navier-like effective stress jump condition is employed to model the presence of the membrane, in which the normal and tangential velocities at the membrane are respectively proportional to the so-called filtrability and slip numbers multiplied by the stresses. Regarding the particular geometry considered here, a characterization of the steady flow for several combinations of constant filtrability and slip numbers shows that the flow morphology is dominantly influenced by the filtrability and exhibits a recirculation region that moves downstream of the body and eventually disappears as this number increases. A linear stability analysis further shows the suppression of vortex shedding as long as large values of the filtrability number are employed. In the control/optimization phase, specific objectives for the macroscopic flow are formulated by adjoint methods. A homogenization-based inverse procedure is proposed to obtain the optimal constrained microscopic geometry from macroscopic objectives, which accounts for fast variations of the filtrability and slip profiles along the membrane. As a test case for the proposed design methodology, a cylindrical membrane is designed to maximize the resulting drag coefficient