Modification of turbulence and stratification of stably stratified turbulent channel flows by finite-size particles

Direct numerical simulations are carried out for thermally stratified turbulent channel flow laden with finite-size particles using an immersed boundary method that was developed for use in nonuniform grids. We consider the stratification effect at friction Richardson numbers ${\text{Ri}}_{\tau }=0$ and 20 with a friction Reynolds number ${\text{Re}}_{\tau }=180$. The particle diameter is chosen as $0.21\delta$ and the global volume fraction of the particles ranges between 0 and $8\%$, where $\delta$ is the channel half gap. Particles are neutrally buoyant or slightly heavier than fluids. We find that neutrally buoyant particles make the internal gravity waves stronger. The vertical motion of the particles is limited to each half of the channel due to the gravity wave, which acts as a wall at the center of the channel. If the particles are slightly heavier than the fluid, the turbulence structures are greatly modified by the particles. The particle movements in the near-wall and core regions are statistically analyzed using particle velocities or residence times, and individual particle movements are presented. The particles stay in the high-speed streak region for a relatively long time and disappear rapidly in the near-wall region when encountering the ejection event of the turbulent flow. Analysis of the vortex structure with the heat flux shows that the particles play a role in suppressing the vertical heat transport. Additional budget analysis of heat flux is performed for a detailed investigation.

Figure 1

Configuration of stably stratified, finite-size particle-laden turbulent channel flow.

Figure 2

Mean shear stresses given by Eq. (20) for the particle-laden turbulent channel flows when ${\text{Ri}}_{\tau }=0$ for (a) $\text{N}\mathrm{\Phi }2$, (b) $\text{N}\mathrm{\Phi }4$, (c) $\text{N}\mathrm{\Phi }8$, and (d) $\text{N}\mathrm{\Phi }2\text{h}$.

Figure 3

Mean velocity profile and relative velocity difference between the particle-free case and the particle-laden case for (a) ${\text{Ri}}_{\tau }=0$ and (b) ${\text{Ri}}_{\tau }=20$. In (a) the results for $\text{N}\mathrm{\Phi }8$ (wall) and Wang et al. (wall) [34], which are comparable to each other, are normalized in the wall units, as indicated in the plot of $({\langle u\rangle }_{\mathrm{PF}}-{\langle u\rangle }_{\mathrm{PL}})/{\langle u\rangle }_{\mathrm{PF}}$.

Figure 4

(a) Mean temperature normalized by $\mathrm{\Delta }T$. (b) Mean temperature gradient normalized by $u_b$ and $T_b$. The wall value corresponds to ${\text{Nu}}_b$. The insets present the profile across the whole channel for $\text{N}\mathrm{\Phi }2\text{h}$ and $\text{S}\mathrm{\Phi }2\text{h}$.

Figure 5

(a) Phase angles [Eq. (23)]. (b) Coherences [Eq. (25)]. (c) Average amplitude of the one-dimensional spectrum of $\langle \widehat{v^{\prime }{\theta }^{\prime }}({\kappa }_x,y=1)\rangle$ at the channel center along the $x$ axis averaged in the $z$ direction. The vertical line in (c) denotes the wave number corresponding to the particle size.

Figure 6

Isosurface of $T=0$ for (a) the particle-free case when ${\text{Ri}}_{\tau }=20$ and (b) the particle-laden case $\text{S}\mathrm{\Phi }8$. The particles presented belong to $0.8<y<1.2$.

Figure 7

Root-mean-square values of fluctuating velocity and the Reynolds shear stress of the fluid for all cases: (a) $u_{\mathrm{rms}}$, (b) $v_{\mathrm{rms}}$, (c) $w_{\mathrm{rms}}$, and (d) $-\langle u^{\prime }{v}^{\prime }\rangle$. The data of Wang et al. $(\diamond )$ [Fig. 9 of 34] and $\mathrm{N}\mathrm{\Phi }8(--)$ are compared.

Figure 8

(a) Vertical heat flux for ${\text{Ri}}_{\tau }=0$ and 20. (b) Contributions from each term in Eq. (26) to the total heat flux for $\text{S}\mathrm{\Phi }8$.

Figure 9

Vortex structures in the near-wall region in $0\le y\le 0.5$ visualized using ${\lambda }_2=-0.01$. The red surface shows ${\theta }^{\prime }{v}^{\prime }=0.015$ and the blue surface ${\theta }^{\prime }{v}^{\prime }=-0.015$ for (a) $\text{S}\mathrm{\Phi }0$ and (b) $\text{S}\mathrm{\Phi }8$.

Figure 10

Mean velocity of the particles for (a) $\text{N}\mathrm{\Phi }2,\text{N}\mathrm{\Phi }8$, and $\text{N}\mathrm{\Phi }2\text{h}$ and (b) $\text{S}\mathrm{\Phi }2,\text{S}\mathrm{\Phi }8$, and $\text{S}\mathrm{\Phi }2\text{h}$. The lines represent the fluids and the symbols represent the particle. All the velocities are normalized by the bulk velocity of each flow.

Figure 11

Root-mean-square values of the fluctuating velocity of the particles for (a) for ${\text{Ri}}_{\tau }=0$ and (b) ${\text{Ri}}_{\tau }=20$. All the quantities are normalized by the bulk velocity of each flow.

Figure 12

Mean angular velocity of the particle in the $z$ direction and the rotational rate of the ambient flow field for (a) $\text{N}\mathrm{\Phi }8$ and $\text{S}\mathrm{\Phi }8$ and (b) $\text{N}\mathrm{\Phi }2\text{h}$ and $\text{S}\mathrm{\Phi }2\text{h}$.

Figure 13

Sampled particle trajectories for the cases $\text{N}\mathrm{\Phi }8$ and $\text{S}\mathrm{\Phi }8$. blue lines are the two-particle trajectories for the case $\text{N}\mathrm{\Phi }8$ and red lines are two-particle trajectories for the case $\text{S}\mathrm{\Phi }8$.

Figure 14

Time- and horizontal-space-averaged local particle volume fraction distribution for all particle-laden cases.

Figure 15

(a) Plan view of the streamwise velocity contour at $y=0.1$ with finite-size particles preferentially located in high-speed streaks for $\text{S}\mathrm{\Phi }8$. (b) Sketch of the methodology to calculate the fluid velocity on the surface of the particle.

Figure 16

Probability density function of $v_p^{\prime }$ in the region $0.133\lesssim y\lesssim 0.167$ for (a) ${\text{Ri}}_{\tau }=0$ and (b) ${\text{Ri}}_{\tau }=20$. (c) The $\langle uv\rangle$ quadrant distribution with the threshold of $-uv/\sqrt{\langle u^{\prime 2}\rangle }\sqrt{\langle v^{\prime 2}\rangle }$ at $y\approx 0.151$ for $\text{N}\mathrm{\Phi }8$ and $\text{S}\mathrm{\Phi }8$. (d) Residence time of the particle in each bin of $y$ given by Eq. (29).

Figure 17

Budgets of $\langle v^{\prime +}{\theta }^{\prime +}\rangle$ in Eq. (30) for (a) $\text{S}\mathrm{\Phi }0$ and (b) $\text{S}\mathrm{\Phi }8$.

Figure 18

Average Nusselt number compared with that obtained by Dhole et al. [67]. Here $\overline{\text{Nu}}$ is the average Nusselt number on the surface of the spherical particle.

Figure 19

Bouncing trajectory of a particle after collision for (a) ${\text{Re}}_p\approx 23$ and (b) ${\text{Re}}_p\approx 50$, with $t^{*}=t/\sqrt{D_p/g}$.

Figure 20

Plot of ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ given by Eqs. (B25) and (B26) for tangential collision.