Role of flow structures on the deposition of low-inertia particles in turbulent pipe flow

We analyze the effect of large-scale coherent structures on the deposition of low-inertia particles in a turbulent pipe flow using extended proper orthogonal decomposition (EPOD) and spectral analysis. We perform direct numerical simulations (DNSs) at two the Reynolds numbers 5300 and 10 300 (based on bulk parameters) with the particles released at the pipe inlet. The equilibrium Eulerian model is employed for calculating particle velocity, and the analysis is limited to particles with Stokes number (based on wall units) less than 1. Increasing the Stokes number increases the energy at small streamwise wavelengths (due to inertial clustering), and the spectral energy peak moves from ${\lambda }_z^{+}\approx 1000$ to ${\lambda }_z^{+}\approx 150$. The spectral peak in the $({\lambda }_z^{+},y^{+})$ plane, where $y^{+}$ is the wall-normal distance, moves from the buffer layer to the logarithmic region. Gravity has a substantial effect on the POD mode shapes. For the downward flow, a second peak appears closer to the center. A new Fukagata-Iwamoto-Kasagi (FIK) identity is derived for the wall deposition rate coefficient (Sherwood number, Sh) and employed to quantify the contributions of the mean and fluctuating velocity and particle concentration fields for different Stokes, Froude, and Reynolds numbers. Modes with azimuthal wave numbers $k_{\theta }$ equal to three or four are found to contribute most to deposition. Application of the developed methodology to higher Reynolds number can elucidate the role of large- and very-large-scale flow structures on particle deposition to the wall. It is well known these structures leave their footprint at the wall but their contribution to deposition is not well understood.

Figure 1

Variation of $\overline{{\text{St}}_k}$ (based on the Kolmogorov timescale ${\tau }_k$) in the wall-normal direction. The dissipation rate used to calculate ${\tau }_k$ is taken from Refs. [43, 44].

Figure 2

(a) Wall-normal particle velocity and (b) the difference between the average streamwise particle and fluid velocities for channel flow at ${\text{Re}}_{\tau }=180$; profiles are compared with the Lagrangian simulation results of Ferry et al. [22]. Solid lines denote the present DNS and $\circ$ the Lagrangian simulations. The blue line denotes ${\text{St}}^{+}=1$ and the red line ${\text{St}}^{+}=3$.

Figure 3

Variation of Sh number and the constituent components ${\text{Sh}}_{\mathrm{FIK}}$ along the length of the pipe for $\text{Re}=5300$, where panel (a) is for ${\text{St}}^{+}=0$, panel (b) is for ${\text{St}}^{+}=1, 1/{\text{Fr}}_z=0$, panel (c) is for ${\text{St}}^{+}=1, {\text{Fr}}_z=0.4$, and panel (d) is for ${\text{St}}^{+}=1, {\text{Fr}}_z=-0.4$.

Figure 4

Bar charts of the nonzero FIK components at $z=15D$ for $\text{Re}=5300$ and $\text{Re}=10300$. (a) Absolute values and (b) values normalized with Sh.

Figure 5

Bar charts of the absolute values of the nonzero inertial FIK components at $z=15D$ for $\text{Re}=5300$ and $\text{Re}=10300$.

Figure 6

Premultiplied spectra of particle concentration for (a), (c) $\text{Re}=5300$ and (b), (d) $\text{Re}=10300$ at $y^{+}=15$ for (a), (b) no-gravity flow, and (c), (d) downward flow or upward flow. The blue line marks 10%, the red line 50%, and the yellow line 75% of the maximum value. In panels (a) and (b) the solid line denotes ${\text{St}}^{+}=0$ (passive scalar) and the dashed line ${\text{St}}^{+}=1$. In panels (c) and (d) the Stokes number is constant (${\text{St}}^{+}=1$), the solid line denotes the downward flow and the dashed line the upward flow.

Figure 7

Instantaneous contours of particle concentration at $\text{Re}=10300$ for (a) ${\text{St}}^{+}=0.0$ and (b) ${\text{St}}^{+}=1$ with $1/{\text{Fr}}_z=0$ (no gravity case).

Figure 8

Premultiplied spectra of particle concentration in the streamwise direction (averaged over the azimuthal direction) for (a), (c) $\text{Re}=5300$ and (b), (d) $\text{Re}=10300$. In panels (a) and (b) ${\text{St}}^{+}=0$ and in panels (c) and (d) ${\text{St}}^{+}=1.0$. Gravity is omitted, $1/{\text{Fr}}_z=0$. The plots are normalized so the maximum energy is unity.

Figure 9

Premultiplied spectra of particle concentration fluctuations in the streamwise direction (averaged over the azimuthal direction) for (a), (c) $\text{Re}=5300$ and (b), (d) $\text{Re}=10300$. In panels (a) and (b) ${\text{Fr}}_z=0.4$ and in panels (c) and (d) ${\text{Fr}}_z=-0.4$. In all plots ${\text{St}}^{+}=1$. The plots are normalized so the maximum energy is unity.

Figure 10

Premultiplied spectra of particle concentration fluctuations in the azimuthal direction (averaged over the streamwise direction) for (a), (c) $\text{Re}=5300$ and (b), (d) $\text{Re}=10300$. In panels (a) and (b) ${\text{St}}^{+}=0$ and in panels (c) and (d) ${\text{St}}^{+}=1.0$. Gravity is omitted, $1/{\text{Fr}}_z=0$. The plots are normalized so the maximum energy is unity.

Figure 11

Premultiplied spectra of particle concentration fluctuations in the azimuthal direction (averaged over the streamwise direction) for (a), (c) $\text{Re}=5300$ and (b), (d) $\text{Re}=10300$. In panels (a) and (b) ${\text{Fr}}_z=0.4$ and in panels (c) and (d) ${\text{Fr}}_z=-0.4$. In all plots ${\text{St}}^{+}=1$. The plots are normalized so the maximum energy is unity.

Figure 12

Eigenvalue distribution of the dominant particle concentration POD mode over $k_{\theta }$ for (a) $\text{Re}=5300$ and (b) $\text{Re}=10300$.

Figure 13

Eigenvalue distribution of the dominant particle concentration POD mode over $k_z$ for (a) $\text{Re}=5300$ and (b) $\text{Re}=10300$.

Figure 14

Eigenvalue distribution of the particle concentration POD modes, $n$, normalized with the total particle concentration variance, integrated over streamwise mode numbers $k_z$ and azimuthal wave numbers $k_{\theta }$, for (a) $\text{Re}=5300$ and (b) $\text{Re}=10300$.

Figure 15

Particle concentration POD modes for (a), (b) $k_z=1,k_{\theta }=4,n=1$ and (c), (d) $k_z=1,k_{\theta }=7,n=1$ at (a), (c) $\text{Re}=5300$ and (b), (d) $\text{Re}=10300$.

Figure 16

Dual-plane contour plots of the particle concentration POD modes at (a) $\text{Re}=5300$ for $k_z=1,k_{\theta }=4,n=1$ and ${\text{St}}^{+}=0, 1/\text{Fr}=0$, (b) ${\text{St}}^{+}=1, 1/\text{Fr}=0$, (c) ${\text{St}}^{+}=1, {\text{Fr}}_z=0.4$, and (d) ${\text{St}}^{+}=1, {\text{Fr}}_z=-0.4$.

Figure 17

Contribution to ${\text{Sh}}_{\mathrm{turb}}$ of the dominant mode ($n=1$) summed over all streamwise wave numbers for (a) $\text{Re}=5300$ and (b) $\text{Re}=10300$.

Figure 18

Contribution to ${\text{Sh}}_{\mathrm{FIK},\mathrm{Ir}}$ of the dominant mode ($n=1$) summed over all streamwise wave numbers for (a) $\text{Re}=5300$ and (b) $\text{Re}=10300$.

Figure 19

Cumulative plot of ${\text{Sh}}_{\mathrm{turb}}$ against $k_{\theta }$ for different $k_z$ values at $\text{Re}=5300$. The solid lines denote $n=1$, and the dashed lines $n=1–50$, where panel (a) is for ${\text{St}}^{+}=0, 1/{\text{Fr}}_z=0$, panel (b) is for ${\text{St}}^{+}=1, 1/{\text{Fr}}_z=0$, panel (c) is for ${\text{St}}^{+}=1, {\text{Fr}}_z=0.4$, and panel (d) is for ${\text{St}}^{+}=1, {\text{Fr}}_z=-0.4$.