Can preferential concentration of finite-size particles in plane Couette turbulence be reproduced with the aid of equilibrium solutions?

This work employs invariant solutions of the Navier-Stokes equations to study the interaction between finite-size particles and near-wall coherent structures. We consider horizontal plane Couette flow and focus on Nagata's upper-branch equilibrium solution at low Reynolds numbers where this solution is linearly stable. When adding a single heavy particle with a diameter equivalent to 2.5 wall units ($\frac{1}{12}$ of the gap width), we observe that the solution remains stable and is essentially unchanged away from the particle. This result demonstrates that it is technically feasible to utilize exact coherent structures in conjunction with particle-resolved direct numerical simulation. While translating in the streamwise direction, the particle migrates laterally under the action of the quasistreamwise vortices until it reaches the region occupied by the low-speed streak, where it attains a periodic state of motion, independent of its initial position. As a result of the ensuing preferential particle location, the time-average streamwise particle velocity differs from the plane-average fluid-phase velocity at the same wall distance as the particle center, as previously observed in experiments and in numerical data for fully turbulent wall-bounded flows. Additional constrained simulations where the particle is maintained at a fixed spanwise position while freely translating in the other two directions reveal the existence of two equilibria located in the low-speed and the high-speed streak, respectively, the former being an unstable point. A parametric study with different particle to fluid density ratios is conducted which shows how inertia affects the spanwise fluctuations of the periodic particle motion. Finally, we discuss a number of potential future investigations of solid particle dynamics which can be conducted with the aid of invariant solutions (exact coherent structures) of the Navier-Stokes equations.

Figure 1

Plane Couette flow geometry illustrating the numerical setup. The physical domain is assumed periodic in the streamwise ($x$) and spanwise ($z$) directions, whereas the $y$ direction is bounded by two walls that move in opposite directions, each with speed $U$. When a solid phase is present, it consists of finite-size spherical particles with diameter $D_p$ and density ${\rho }_p$. Gravity $\mathbf{g}$ acts in the negative $y$ direction.

Figure 2

State-space diagram for Nagata's solution in terms of the energy dissipation rate ${\mathcal{E}}_D$ and the Reynolds number: domain ${\mathit{\Omega }}_1$ with $({\lambda }_x,{\lambda }_z)=(6h,3h)$ and domain ${\mathit{\Omega }}_2$ with $({\lambda }_x,{\lambda }_z)=(12h,6h)$, where ${\lambda }_x$ and ${\lambda }_z$ are the fundamental wavelengths in the streamwise and spanwise directions, respectively. The blue (red) shading indicates stability (instability) of the solution, as determined in Sec. 3b. The blue dot on the upper branch of the solution in box ${\mathit{\Omega }}_2$ marks the selected parameter point for the present study.

Figure 3

Example of the stability analysis showing an unstable and a stable equilibrium solution. In (a) we see that for domain ${\mathit{\Omega }}_1$ and $\text{Re}=190.62$ the initial disturbances are amplified with time and that after approximately $200{\tau }_b$ the laminar base flow is recovered. In (b) the initial disturbance in domain ${\mathit{\Omega }}_2$ with $\text{Re}=132.25$ is continuously damped and the unperturbed case is recovered for $t>1000{\tau }_b$.

Figure 4

Upper branch of Nagata's solutions at $\text{Re}=132.25$ and $({\lambda }_x,{\lambda }_z)=(12h,6h)$. (a) Isosurfaces of $u=\text{min}\left[u\right(x,y=0,z\left)\right]$ (green) and the $Q$ criterion of Hunt et al. [36] with $Q=0.7\text{max}\left(Q\right)$, colored according to the sign of the streamwise vorticity, i.e., ${\omega }_x<0$ (blue) and ${\omega }_x>0$ (red). (b) Mean streamwise velocity normalized by the wall speed. (c) Root mean square velocities in wall units: ${\langle u_f^{\prime }{u}_f^{\prime }\rangle }_{xz}^{1/2}/u_{\tau }$ ($—$), ${\langle v_f^{\prime }{v}_f^{\prime }\rangle }_{xz}^{1/2}/u_{\tau }$ ($\sout{ ◻ }$), and ${\langle w_f^{\prime }{w}_f^{\prime }\rangle }_{xz}^{1/2}/u_{\tau }$ ($\sout{ △ }$).

Figure 5

(a) Time evolution of the difference between the box-average kinetic energy in the single-phase case (${\langle K_{sp}^{fd}\rangle }_t$) and in run S10 ($K^{fd}$). (b) Zoom of the fluctuations of the box-average kinetic energy ${\left(K^{fd}\right)}^{\prime }$ around its mean value in the periodic regime, shown for the final few cycles of case S10.

Figure 6

Isocontours of fluctuations of the streamwise fluid velocity in an $(x,z)$ plane that intersects the particle centroid. The isocontours evidence the presence of a low-speed streak centered around $z/h=3$. Note that the depicted flow field corresponds to the unperturbed flow. The yellow circle denotes the initial particle position. The dotted line indicates the trajectory of the particle over the course of the simulation.

Figure 7

Spanwise particle position over time in bulk units for run S10. The inset shows a close-up of the data in the asymptotic regime, when the motion is periodic with period $T_p=3.93{\tau }_b$ and an amplitude of $0.09h$.

Figure 8

Same graph as in Fig. 6, but showing the initial positions (black circles) of the suspended particles in the multiparticle case M10, as well as the particle positions after an elapsed time of $120{\tau }_b$ (yellow circles).

Figure 9

Spanwise particle position $z_p$ over time in bulk units for the multiparticle case M10. Each line corresponds to a different particle.

Figure 10

Time-average spanwise force for the last cycle in the constrained particle case C10, plotted as a function of the imposed spanwise position, i.e., one datum per particle.

Figure 11

Single-particle simulations at various solid to fluid density ratios. The data are for the asymptotic regime with fully developed periodic particle motion. (a) Spanwise particle position as a function of the streamwise position. (b) Spanwise particle velocity as a function of the streamwise position. The color code in both graphs refers to each run S04–S80 in Table 3, where the arrow indicates the direction of increase of ${\rho }_p/{\rho }_f$. The dashed curve in (b) shows the spanwise fluid velocity of the unperturbed flow field at the mean low-speed streak location and at a wall distance equivalent to the particle's centroid.

Figure 12

Amplitude of (a) the spanwise particle motion and (b) the spanwise particle velocity as a function of density ratio for cases S04–S80.

Figure 13

Time evolution of the box-average kinetic energy of the fluctuations for Nagata's solution with $({\lambda }_x,{\lambda }_z)=(12h,6h)$ and $\text{Re}=132.25$ when integrated using a pseudospectral time stepper as described in Sec. 2b2 ($—$) and with the second-order finite-difference solver as described in Sec. 2b3 with a uniform grid featuring widths of $(\mathrm{\Delta }{x}^{+},\mathrm{\Delta }{y}^{+},\mathrm{\Delta }{z}^{+})=(0.50,0.25,0.50)$, starting from a spectrally interpolated solution at time $t=0$ ($‐‐‐$).

Figure 14

Time evolution of the box-average kinetic energy of the fluctuations $K^{fd}$ in the baseline case S10 ($—$) and after grid refinement, S10-F ($–$). The inset shows a close-up over the last few cycles of the latter run.

Figure 15

Data over one cycle in the asymptotic regime in case S10 ($—$) and using the refined grid of S10-F ($–$) showing (a) the spanwise particle position and (b) the spanwise particle velocity component. In (b) the blue open circles represent the data from case S10-F scaled by the ratio of the two amplitudes.