On the damped oscillations of an elastic quasi-circular membrane in a two-dimensional incompressible fluid

We propose a procedure - partly analytical and partly numerical - to find the frequency and the damping rate of the small-amplitude oscillations of a massless elastic capsule immersed in a two-dimensional viscous incompressible fluid. The unsteady Stokes equations for the stream function are decomposed onto normal modes for the angular and temporal variables, leading to a fourth-order linear ordinary differential equation in the radial variable. The forcing terms are dictated by the properties of the membrane, and result into jump conditions at the interface between the internal and external media. The equation can be solved numerically, and an excellent agreement is found with a fully-computational approach we developed in parallel. Comparisons are also shown with the results available in the scientific literature for drops, and a model based on the concept of embarked fluid is presented, which allows for a good representation of the results and a consistent interpretation of the underlying physics.Comment: in press on JF

Kazantsev dynamo in turbulent compressible flows

We consider the kinematic fluctuation dynamo problem in a flow that is random, white-in-time, with both solenoidal and potential components. This model is a generalization of the well-studied Kazantsev model. If both the solenoidal and potential parts have the same scaling exponent, then, as the compressibility of the flow increases, the growth rate decreases but remains positive. If the scaling exponents for the solenoidal and potential parts differ, in particular if they correspond to typical Kolmogorov and Burgers values, we again find that an increase in compressibility slows down the growth rate but does not turn it off. The slow down is, however, weaker and the critical magnetic Reynolds number is lower than when both the solenoidal and potential components display the Kolmogorov scaling. Intriguingly, we find that there exist cases, when the potential part is smoother than the solenoidal part, for which an increase in compressibility increases the growth rate. We also find that the critical value of the scaling exponent above which a dynamo is seen is unity irrespective of the compressibility. Finally, we realize that the dimension $d = 3$ is special, since for all other values of $d$ the critical exponent is higher and depends on the compressibility.Comment: 12 pages, 6 figure

Optimal transient growth in an incompressible flow past a backward-slanted step

With the aim of providing a first step in the quest for a reduction of the aerodynamic drag on the rear-end of a car, we study the phenomena of separation and reattachment of an incompressible flow focusing on a specific aerodynamic geometry, namely a backward-slanted step at 25 degrees of inclination. The ensuing recirculation bubble provides the basis for an analytical and numerical investigation of streamwise-streak generation, lift-up effect, and turbulent-wake and Kelvin-Helmholtz instabilities. A linear stability analysis is performed, and an optimal control problem with a steady volumic forcing is tackled by means of variational formulation, adjoint method, penalization scheme and orthogonalization algorithm. Dealing with the transient growth of spanwise-periodic perturbations and inspired by the need of physically-realizable disturbances, we finally provide a procedure attaining a kinetic-energy maximal gain of the order of one million with respect to the power introduced by the external forcing.Comment: 17 figure

Renormalized transport of inertial particles in surface flows

Surface transport of inertial particles is investigated by means of the perturbative approach, introduced by Maxey (J. Fluid Mech. 174, 441 (1987)), which is valid in the case the deflections induced on the particle trajectories by the fluid flow can be considered small. We consider a class of compressible random velocity fields, in which the effect of recirculations is modelled by an oscillatory component in the Eulerian time correlation profile. The main issue we address here is whether fluid velocity fluctuations, in particular the effect of recirculation, may produce nontrivial corrections to the streaming particle velocity. Our result is that a small (large) degree of recirculation is associated with a decrease (increase) of streaming with respect to a quiescent fluid. The presence of this effect is confirmed numerically, away from the perturbative limit. Our approach also allows us to calculate the explicit expression for the eddy diffusivity, and to compare the efficiency of diffusive and ballistic transport.Comment: 18 pages, 13 figures, submitted to JF

Does multifractal theory of turbulence have logarithms in the scaling relations?

The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random multiplicative model of turbulence and calculating subdominant terms, we explain here why such corrections cannot be present.Comment: 7 pages, 1 figur

Point-source inertial particle dispersion

The dispersion of inertial particles continuously emitted from a point source is analytically investigated in the limit of small inertia. Our focus is on the evolution equation of the particle joint probability density function p(x,v,t), x and v being the particle position and velocity, respectively. For finite inertia, position and velocity variables are coupled, with the result that p(x,v,t) can be determined by solving a partial differential equation in a 2d-dimensional space, d being the physical-space dimensionality. For small inertia, (x,v)-variables decouple and the determination of p(x,v,t) is reduced to solve a system of two standard forced advection-diffusion equations in the space variable x. The latter equations are derived here from first principles, i.e. from the well-known Lagrangian evolution equations for position and particle velocity.Comment: 10 pages, submitted to JF