Inelastic accretion of inertial particles by a towed sphere

The problem of accretion of small particles by a sphere embedded in a mean flow is studied in the case where the particles undergo inelastic collisions with the solid object. The collision efficiency, which gives the flux of particles experiencing at least one bounce on the sphere, is found to depend upon the sphere Reynolds number only through the value of the critical Stokes number below which no collision occurs. In the absence of molecular diffusion, it is demonstrated that multiple bounces do not provide enough energy dissipation for the particles to stick to the surface within a finite time. This excludes the possibility of any kind of inelastic collapse, so that determining an accretion efficiency requires modeling more precisely particle-surface microphysical interactions. A straightforward choice is to assume that the particles stick when their kinetic energy at impact is below a threshold. In this view, numerical simulations are performed to describe the statistics of impact velocities at various values of the Reynolds number. Successive bounces are shown to enhance accretion. These results are put together to provide a general qualitative picture on how the accretion efficiency depends upon the nondimensional parameters of the problem.

Figure 1

Sketch of the various regimes occurring in the flow past an immobile sphere as a function of the Reynolds number $\text{Re}=Ud/\nu$. The color volumes are a cartoon of vorticity isosurfaces. The red arrows correspond to the values considered in the numerical simulations of this work.

Figure 2

Streamlines corresponding to the two limiting axisymmetric cases of a creeping (Stokes) flow (left) and a potential inviscid (Euler) flow (right). Two particle trajectories have been represented in both cases as bold curves. One which collides with the sphere (right-most, in red), and one which does not (left-most, magenta).

Figure 3

Collision efficiency ${\mathcal{E}}_{\mathrm{coll}}$ as a function of $\text{St}$ for the various Reynolds numbers considered here, including $\text{Re}=0$ (Stokes creeping flow) and $\text{Re}=\infty$ (Euler inviscid flow).

Figure 4

Collision efficiency ${\mathcal{E}}_{\mathrm{coll}}$ as a function of $\text{St}/{\text{St}}_{\mathrm{c}}-1,$ where ${\text{St}}_{\mathrm{c}}$ is the critical Stokes number below which no collisions happen. The data of the various simulations are represented as filled symbols, as labeled. The results of Homann et al. [22] at comparable Reynolds numbers are shown as empty symbols. The solid curve corresponds to the fitting master curve Eq. (10). Inset: Critical Stokes number as a function of the Reynolds number for the various runs. Equation (3) proposed by Slinn [20] is shown as a solid curve. The two horizontal lines are the two asymptotes (Stokes flow ${\text{St}}_{\mathrm{c}}=0.605$ and Euler flow ${\text{St}}_{\mathrm{c}}=1/24)$.

Figure 5

Log-log plot of $-log\left({\mathcal{E}}_{\mathrm{coll}}\right)$ (crosses) in the case of the Euler flow as a function of $\text{St}/{\text{St}}_{\mathrm{c}}-1,$ where ${\text{St}}_{\mathrm{c}}=1/24$ is the critical Stokes number. The line shows a behavior $-\text{log}\left({\mathcal{E}}_{\mathrm{coll}}\right)\propto {(\text{St}-{\text{St}}_{\mathrm{c}})}^{-1/2}$.

Figure 6

Average concentration in the $y=0$ plane of particles with Stokes number $\text{St}={\tau }_{\mathrm{p}}U/d=1$ in a $\text{Re}=0$ flow with an absorbing boundary condition at the surface of the sphere (left) and when they bounce (right) with a coefficient of restitution $e=0.5$.

Figure 7

Examples of particle trajectories for $\text{St}=10$ and three different restitution coefficients $(e=0.5$, 0.75, and 1, as labelled) in the Stokes flow. The particles are injected at the same initial value of $\rho$ close to the symmetry axis and experience multiple collisions with the sphere. The left panel shows their location in the $(\rho ,z)$ plane and the right panel is the distance to the boundary surface as a function of time.

Figure 8

Two trajectories of the system Eq. (13) with $\alpha =2$ associated to values of the Stokes number $S_0$ below (blue, solid line) and above (red, dashed line) the critical Stokes number $S_{★}(\alpha =2)\approx 1.18$. Inset: critical Stokes number $S_{★}$ as a function of the fluid velocity exponent $\alpha$.

Figure 9

Phase portraits of the $(z_{\mathrm{p}},{\dot{z}}_{\mathrm{p}})$ dynamics for $\alpha =2$ (left) and $\alpha =1/2$ (right). When $\alpha >1$ and for any value of the Stokes number $S_0$, there exists a critical trajectory (represented as a bold red line) such that all trajectories above touch $z_{\mathrm{p}}=0$ in a finite time with a finite velocity and all those below are tending asymptotically to the origin. When $\alpha <1$, there is no such critical trajectory and all initial conditions lead to touch $z_{\mathrm{p}}=0$ in a finite time.

Figure 10

Average of the modulus of the impact velocity $v_1=\langle v_1^{\mathrm{coll}}\rangle$ at the first collision, as a function of the Stokes number for the various Reynolds numbers, including the limiting cases of the Stokes creeping flow and the inviscid potential flow.

Figure 11

Deviations from the inviscid potential case of the average impact velocity at the first collision. They collapse on the top of each other when represented as a function of $(\text{St}-{\text{St}}_c)\sqrt{\text{Re}}$. The black line shows a slope $-3/4$.

Figure 12

$v_1^{\mathrm{coll}}$ at the first collision as a function of the injection location ${\rho }_0$ for $\text{St}=2$ and the various Reynolds numbers that we have considered. Inset: Probability density function of the first impact velocity $v_1^{\mathrm{coll}}$ for $\text{St}=2$ and various flows (Stokes flow, $\text{Re}=100$, and Euler flow).

Figure 13

Probability density function of the impact velocity $v^{\mathrm{coll}}$ for $\text{Re}=100,\text{St}=2$ and $e=1$. The distribution is multimodal, with the various modes associated to successive bounces of the particles. The vertical dashed lines show the averages impact velocities $\langle v_n^{\mathrm{coll}}\rangle$ at the $n\mathrm{th}$ collision.

Figure 14

Average of the impact velocity $\langle v_n^{\mathrm{coll}}\rangle$ at the $n\mathrm{th}$ bounce as a function of $n$ for $\text{St}=2,e=1$, and the various Reynolds numbers that we have considered.

Figure 15

Minimal value of the impact parameter $\gamma ={min}_n{v}_n^{\mathrm{coll}}/\left(U{\text{St}}^{1/4}\right)$ (colored background ) as a function of the particles Stokes number $\text{St}$ and of the square ${\rho }_0^2$ of their initial distance to the symmetry axis in the Stokes creeping flow and for a restitution coefficient $e=1$. The black bold lines correspond to the critical curves delimiting the set of parameters for which particles are experiencing at least $n$ collisions (see text). The two vertical lines correspond to the cut shown in Fig. 16.

Figure 16

Impact parameter $\gamma$ for the Stokes flow and elastic particles $\left(e=1\right)$ for two values of the Stokes number $(\text{St}=1.5$ and $\text{St}=4)$. The horizontal dashed line correspond to a specific choice ${\gamma }_{★}=0.1$ of the accretion critical parameter. All particles such that $\gamma <{\gamma }_{★}$ stick to the sphere. The inset represents for the same settings the accretion efficiency associated to this specific value of ${\gamma }_{★}$, as a function of the Stokes number.

Figure 17

Maximum number of collisions experienced by particles in the Stokes flow, as a function of the Stokes number $\text{St}$ and of the restitution coefficient $e$. The black lines show the dependence upon $e$ of the critical Stokes number ${\text{St}}_c^{\left(n\right)}$ above which particles experience at least $n$ bounces.