We obtain, by extensive direct numerical simulations, trajectories of heavy inertial particles in two-dimensional, statistically steady, homogeneous, and isotropic turbulent flows, with friction. We show that the probability distribution function $ma
thcal{P}(kappa)$, of the trajectory curvature $kappa$, is such that, as $kappa to infty$, $mathcal{P}(kappa) sim kappa^{-h_{rm r}}$, with $h_{rm r} = 2.07 pm 0.09$. The exponent $h_{rm r}$ is universal, insofar as it is independent of the Stokes number ($rm{St}$) and the energy-injection wave number. We show that this exponent lies within error bars of their counterparts for trajectories of Lagrangian tracers. We demonstrate that the complexity of heavy-particle trajectories can be characterized by the number $N_{rm I}(t,{rm St})$ of inflection points (up until time $t$) in the trajectory and $n_{rm I} ({rm St}) equiv lim_{ttoinfty} frac{N_{rm I} (t,{rm St})}{t} sim {rm St}^{-Delta}$, where the exponent $Delta = 0.33 pm0.02$ is also universal.
As a test bed for the growth of protoplanetary bodies in a turbulent circumstellar disk we examine the fate of a boulder using direct numerical simulations of particle seeded gas flowing around it. We provide an accurate description of the flow by im
posing no-slip and non-penetrating boundary conditions on the boulder surface using the immersed boundary method pioneered by Peskin (2002). Advected by the turbulent disk flow, the dust grains collide with the boulder and we compute the probability density function (PDF) of the normal component of the collisional velocity. Through this examination of the statistics of collisional velocities we test the recently developed concept of collisional fusion which provides a physical basis for a range of collisional velocities exhibiting perfect sticking. A boulder can then grow sufficiently rapidly to settle into a Keplerian orbit on disk evolution time scales.
Shear thickening appears as an increase of the viscosity of a dense suspension with the shear rate, sometimes sudden and violent at high volume fraction. Its origin for noncolloidal suspension with non-negligible inertial effects is still debated. He
re we consider a simple shear flow and demonstrate that fluid inertia causes a strong microstructure anisotropy that results in the formation of a shadow region with no relative flux of particles. We show that shear thickening at finite inertia can be explained as an increase of the effective volume fraction when considering the dynamically excluded volume due to these shadow regions
We consider mean-field dynamo models with fluctuating alpha effect, both with and without shear. The alpha effect is chosen to be Gaussian white noise with zero mean and given covariance. We show analytically that the mean magnetic field does not gro
w, but, in an infinitely large domain, the mean-squared magnetic field shows exponential growth of the fastest growing mode at a rate proportional to the shear rate, which agrees with earlier numerical results of Yousef et al (2008) and recent analytical treatment by Heinemann et al (2011) who use a method different from ours. In the absence of shear, an incoherent alpha^2 dynamo may also be possible. We further show by explicit calculation of the growth rate of third and fourth order moments of the magnetic field that the probability density function of the mean magnetic field generated by this dynamo is non-Gaussian.
We present direct numerical simulations of turbulent channel flow with passive Lagrangian polymers. To understand the polymer behavior we investigate the behavior of infinitesimal line elements and calculate the probability distribution function (PDF
) of finite-time Lyapunov exponents and from them the corresponding Cramers function for the channel flow. We study the statistics of polymer elongation for both the Oldroyd-B model (for Weissenberg number $Wi <1$) and the FENE model. We use the location of the minima of the Cramers function to define the Weissenberg number precisely such that we observe coil-stretch transition at $Wiapprox1$. We find agreement with earlier analytical predictions for PDF of polymer extensions made by Balkovsky, Fouxon and Lebedev [Phys. Rev. Lett., 84, 4765 (2000).] for linear polymers (Oldroyd-B model) with $Wi<1$ and by Chertkov [Phys. Rev. Lett., 84, 4761 (2000).] for nonlinear FENE-P model of polymers. For $Wi>1$ (FENE model) the polymer are significantly more stretched near the wall than at the center of the channel where the flow is closer to homogenous isotropic turbulence. Furthermore near the wall the polymers show a strong tendency to orient along the stream-wise direction of the flow but near the centerline the statistics of orientation of the polymers is consistent with analogous results obtained recently in homogeneous and isotropic flows.
Simulations of stochastically forced shear-flow turbulence in a shearing-periodic domain are used to study the spontaneous generation of large-scale flow patterns in the direction perpendicular to the plane of the shear. Based on an analysis of the r
esulting large-scale velocity correlations it is argued that the mechanism behind this phenomenon could be the mean-vorticity dynamo effect pioneered by Elperin, Kleeorin, and Rogachevskii in 2003 (Phys. Rev. E 68, 016311). This effect is based on the anisotropy of the eddy viscosity tensor. One of its components may be able to replenish cross-stream mean flows by acting upon the streamwise component of the mean flow. Shear, in turn, closes the loop by acting upon the cross-stream mean flow to produce stronger streamwise mean flows. The diagonal component of the eddy viscosity is found to be of the order of the rms turbulent velocity divided by the wavenumber of the energy-carrying eddies.
We study the dependence of turbulent transport coefficients, such as the components of the $alpha$ tensor ($alpha_{ij}$) and the turbulent magnetic diffusivity tensor ($eta_{ij}$), on shear and magnetic Reynolds number in the presence of helical forc
ing. We use three-dimensional direct numerical simulations with periodic boundary conditions and measure the turbulent transport coefficients using the kinematic test field method. In all cases the magnetic Prandtl number is taken as unity. We find that with increasing shear the diagonal components of $alpha_{ij}$ quench, whereas those of $eta_{ij}$ increase. The antisymmetric parts of both tensors increase with increasing shear. We also propose a simple expression for the turbulent pumping velocity (or $gamma$ effect). This pumping velocity is proportional to the kinetic helicity of the turbulence and the vorticity of the mean flow. For negative helicity, i.e. for a positive trace of $alpha_{ij}$, it points in the direction of the mean vorticity, i.e. perpendicular to the plane of the shear flow. Our simulations support this expression for low shear and magnetic Reynolds number. The transport coefficients depend on the wavenumber of the mean flow in a Lorentzian fashion, just as for non-shearing turbulence.
