We investigate the effect of inertial particles dispersed in a circular patch of finite radius on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where no unstable modes exist, we show that the feedback force from the particles triggers a novel instability. The mechanisms driving the instability are characterized using linear stability analysis for weakly inertial particles and further validated against Eulerian–Lagrangian simulations. We show that the particle-laden vortex is always unstable if the mass loading $M>0$. Surprisingly, even non-inertial particles destabilize the vortex by a mechanism analogous to the centrifugal Rayleigh–Taylor instability in radially stratified vortex with density jump. We identify a critical mass loading above which an eigenmode $m$ becomes unstable. This critical mass loading drops to zero as $m$ increases. When particles are inertial, modes that fall below the critical mass loading become unstable, whereas modes above it remain unstable but with lower growth rates compared with the non-inertial case. Comparison with Eulerian–Lagrangian simulations shows that growth rates computed from simulations match well the theoretical predictions. Past the linear stage, we observe the emergence of high-wavenumber modes that turn into spiralling arms of concentrated particles emanating out of the core, while regions of particle-free flow are sucked inward. The vorticity field displays a similar pattern which leads to the breakdown of the initial Rankine structure. This novel instability for a dusty vortex highlights how the feedback force from the disperse phase can induce the breakdown of an otherwise resilient vortical structure.

We investigate the effect of inertial particles on the stability and decay of a prototypical vortex tube, represented by a two-dimensional Lamb–Oseen vortex. In the absence of particles, the strong stability of this flow makes it resilient to perturbations, whereby vorticity and enstrophy decay at a slow rate controlled by viscosity. Using Eulerian–Lagrangian simulations, we show that the dispersion of semidilute inertial particles accelerates the decay of the vortex tube by orders of magnitude. In this work, mass loading is unity, ensuring that the fluid and particle phases are tightly coupled. Particle inertia and vortex strength are varied to yield Stokes numbers 0.1–0.4 and circulation Reynolds numbers 800–5000. Preferential concentration causes these inertial particles to be ejected from the vortex core forming a ring-shaped cluster and a void fraction bubble that expand outwards. The outward migration of the particles causes a flattening of the vorticity distribution, which enhances the decay of the vortex. The latter is further accelerated by small-scale clustering that causes enstrophy to grow, in contrast with the monotonic decay of enstrophy in single-phase two-dimensional vortices. These dynamics unfold on a time scale that is set by preferential concentration and is two orders of magnitude lower than the viscous time scale. Increasing particle inertia causes a faster decay of the vortex. This work shows that the injection of inertial particles could provide an effective strategy for the control and suppression of resilient vortex tubes.

In the flamelet regime of turbulent premixed combustion the enhancement in the burning rates originates primarily from surface wrinkling. In this work we investigate the Reynolds number dependence of burning rates of spherical turbulent premixed methane/air flames in decaying isotropic turbulence with direct numerical simulations. Several simulations are performed by varying the Reynolds number, while keeping the Karlovitz number the same, and the temporal evolution of the flame surface is compared across cases by combining the probability density function of the radial distance of the flame surface from the origin with the surface density function formalism. Because the mean area of the wrinkled flame surface normalized by the area of a sphere with radius equal to the mean flame radius is proportional to the product of the turbulent flame brush thickness and peak surface density within the brush, the temporal evolution of the brush and peak surface density are investigated separately. The brush thickness is shown to scale with the integral scale of the flow, evolving due to decaying velocity fluctuations and stretch. When normalized by the integral scale, the wrinkling scale defined as the inverse of the peak surface density is shown to scale with Reynolds number across simulations and as turbulence decays. As a result, the area ratio and the burning rate are found to increase as ${Re}_{\lambda }^{1.13}$, in agreement with recent experiments on spherical turbulent premixed flames. We observe that the area ratio does not vary with turbulent intensity when holding the Reynolds number constant.

In this study, we address the modification of sheared turbulence by dispersed inertial particles. The preferential sampling of the straining regions of the flow by inertial particles in turbulence leads to an inhomogeneous distribution of particles. The strong gravitational loading exerted by the highly concentrated regions results in anisotropic alteration of turbulence at small scales in the direction of gravity. These effects are investigated in a rapid distortion theory (RDT) extended for two-way coupled particle-laden flows. To make the analysis tractable, we assume that particles have small but non-zero inertia. In the classical results for single-phase flows, the RDT assumption of fast shearing compared to the turbulence time scales leads to the distortion and shear-induced production of turbulence. In particle-laden turbulence, the coupling between the two phases under rapid shearing induces number density fluctuations that convert gravitational potential energy to turbulent kinetic energy and modulate the turbulence spectrum in a manner that increases with mass loading. Turbulence statistics obtained from RDT are compared with Euler–Lagrange simulations of homogeneously sheared particle-laden turbulence.

Particle-laden flows of sedimenting solid particles or droplets in a carrier gas have strong inter-phase coupling. Even at low particle volume fractions, the two-way coupling can be significant due to the large particle to gas density ratio. In this semi-dilute regime, the slip velocity between phases leads to sustained clustering that strongly modulates the overall flow. The analysis of perturbations in homogeneous shear reveals the process by which clusters form: (i) the preferential concentration of inertial particles in the stretching regions of the flow leads to the formation of highly concentrated particle sheets, (ii) the thickness of the latter is controlled by particle-trajectory crossing, which causes a local dispersion of particles, (iii) a transverse Rayleigh–Taylor instability, aided by the shear-induced rotation of the particle sheets towards the gravity normal direction, breaks the planar structure into smaller clusters. Simulations in the Euler–Lagrange formalism are compared to Euler–Euler simulations with the two-fluid and anisotropic-Gaussian methods. It is found that the two-fluid method is unable to capture the particle dispersion due to particle-trajectory crossing and leads instead to the formation of discontinuities. These are removed with the anisotropic-Gaussian method which derives from a kinetic approach with particle-trajectory crossing in mind.

Simulations of homogeneously sheared turbulence (HST) are conducted until a universal self-similar state is established at the long non-dimensional time $\unicode[STIX]{x1D6E4}t=20$, where $\unicode[STIX]{x1D6E4}$ is the shear rate. The simulations are enabled by a new robust and discretely conservative algorithm. The method solves the governing equations in physical space using the so-called shear-periodic boundary conditions. Convection by the mean homogeneous shear flow is treated implicitly in a split step approach. An iterative Crank–Nicolson time integrator is chosen for robustness and stability. The numerical strategy captures without distortion the Kelvin modes, rotating waves that are fundamental to homogeneously sheared flows and are at the core of rapid distortion theory. Three direct numerical simulations of HST with the initial Taylor scale Reynolds number $Re_{\unicode[STIX]{x1D706}0}=29$ and shear numbers of $S_{0}^{\ast }=\unicode[STIX]{x1D6E4}q^{2}/\unicode[STIX]{x1D716}=3$, 15 and 27 are performed on a $2048\times 1024\times 1024$ grid. Here, $\unicode[STIX]{x1D716}$ is the dissipation rate and $1/2q^{2}$ is the turbulent kinetic energy. The long integration time considered allows the establishment of a self-similar state observed in experiments but often absent from simulations conducted over shorter times. The asymptotic state appears to be universal with a long time production to dissipation rate ${\mathcal{P}}/\unicode[STIX]{x1D716}\sim 1.5$ and shear number $S^{\ast }\sim 10$ in agreement with experiments. While the small scales exhibit strong anisotropy increasing with initial shear number, the skewness of the transverse velocity derivative decreases with increasing Reynolds number.

We examine the linear stability of a homogeneous gas–solid suspension of small Stokes number particles, with a moderate mass loading, subject to a simple shear flow. The modulation of the gravitational force exerted on the suspension, due to preferential concentration of particles in regions of low vorticity, in response to an imposed velocity perturbation, can lead to an algebraic instability. Since the fastest growing modes have wavelengths small compared with the characteristic length scale ($U_{g}/{\it\Gamma}$) and oscillate with frequencies large compared with ${\it\Gamma}$, $U_{g}$ being the settling velocity and ${\it\Gamma}$ the shear rate, we apply the WKB method, a multiple scale technique. This analysis reveals the existence of a number density mode which travels due to the settling of the particles and a momentum mode which travels due to the cross-streamline momentum transport caused by settling. These modes are coupled at a turning point which occurs when the wavevector is nearly horizontal and the most amplified perturbations are those in which a momentum wave upstream of the turning point creates a downstream number density wave. The particle number density perturbations reach a finite, but large amplitude that persists after the wave becomes aligned with the velocity gradient. The growth of the amplitude of particle concentration and fluid velocity disturbances is characterised as a function of the wavenumber and Reynolds number ($\mathit{Re}=U_{g}^{2}/{\it\Gamma}{\it\nu}$) using both asymptotic theory and a numerical solution of the linearised equations.