Drag enhancement in a dusty Kolmogorov flow

Particles suspended in a fluid exert feedback forces that can significantly impact the flow, altering the turbulent drag and velocity fluctuations. We study flow modulation induced by small spherical particles heavier than the carrier fluid in the framework of an Eulerian two-way coupled model, where particles are represented by a continuum density transported by a compressible velocity field, exchanging momentum with the fluid phase. We implement the model in direct numerical simulations of the turbulent Kolmogorov flow, a simplified setting allowing for studying the momentum balance and the turbulent drag in the absence of boundaries. We show that the amplitude of the mean flow and the turbulence intensity are reduced by increasing particle mass loading with the consequent enhancement of the friction coefficient. Surprisingly, turbulence suppression is stronger for particles of smaller inertia. We understand such a result by mapping the equations for dusty flow, in the limit of vanishing inertia, to a Newtonian flow with an effective forcing reduced by the increase in fluid density due to the presence of particles. We also discuss the negative feedback produced by turbophoresis, which mitigates the effects of particles, especially with larger inertia, on the turbulent flow.

Figure 1

Visualization of two-dimensional sections in the plane $(x,z)$ (at fixed $y=L/2$) of the particle density field $\theta$ (top), and longitudinal velocity field $u_x$ (bottom) normalized with the amplitude of the mean flow $U$. Simulations refer to $\tau =0.34$ and ${\mathrm{\Phi }}_m$ as labeled.

Figure 2

Averaged profiles and amplitudes of the longitudinal fluid velocity. (a) Mean velocity profile $\overline{u_x}\left(z\right)$ for different mass loading ${\mathrm{\Phi }}_m=(0.0,0.4,1.0)$ and fixed $\tau =0.58$. (b) Amplitude of the mean flow $U$ as a function of ${\mathrm{\Phi }}_m$ for different Stokes time as in label.

Figure 3

Profiles of the mean longitudinal flow $\overline{u_x}\left(z\right))$ (red circles and solid line) and the square velocity fluctuations $\overline{|{\mathit{u}}^{\prime }{|}^2}\left(z\right)$ (blue squares and dashed line), for a simulation of the particle-laden flow with $\tau =0.1$, ${\mathrm{\Phi }}_m=1.0$ (symbols) and a simulation of a pure fluid with rescaled forcing amplitude $F^{\prime }=F/(1+{\mathrm{\Phi }}_m)$ (black lines, data from Ref. [44]).

Figure 4

(a) RMS fluid velocity fluctuations $u_{\mathrm{rms}}^{\prime }$ as a function of ${\mathrm{\Phi }}_m$. In the inset velocity fluctuations are normalized with the mean flow. (b) RMS particle density fluctuations ${\theta }_{\mathrm{rms}}^{\prime }$ as a function of St.

Figure 5

(a) Mean particle density profile $\overline{\theta }\left(z\right)$ for different values of mass loading ${\mathrm{\Phi }}_m=(0.0,0.4,1.0)$ and fixed Stokes time $\tau =0.58$. (b) Amplitude $A$ of the spatial modulation of the density profile $\overline{\theta }\left(z\right)=1+Acos\left(2Kz\right)$, as a function of the Stokes number $St$ for different values of ${\mathrm{\Phi }}_m$.

Figure 6

Momentum budget. Mean profile (a) of the Reynolds stress term and (b) of the momentum exchange term for different mass loadings as in label, with $\tau =0.58$. (c) shows the momentum budget for the amplitudes divided into Reynolds stress term (filled symbols) and exchange term (empty symbols), as a function of the mass loading ${\mathrm{\Phi }}_m$ for different values of Stokes times as in label.

Figure 7

Friction and stress coefficients. (a) Friction factor $f$ (filled symbols) as a function of Re for all the parameter configurations (${\mathrm{\Phi }}_m,\tau$) as in legend. The black pentagons are the values of $f$ in the absence of particles (${\mathrm{\Phi }}_m=0$) and the black continuous line is $f=f_0+b/\text{Re}$ with $f_0=0.124$ and $b=5.75$ [44]. Dashed line represents the curve $f=F/\left({\nu }^2{K}^3{\text{Re}}^2\right)$. Blue asterisk corresponds to the simulation with imposed uniform density ($\theta =1$) at ${\mathrm{\Phi }}_m=1$ and $\tau =0.58$ (same parameters of run C3). (b) Stress coefficient $\sigma$ (solid curves) and exchange coefficient $\chi$ (dotted curves).

Figure 8

Mean particle density profile $\overline{\theta }\left(z\right)$ for different values of Stokes time (see legend) obtained with the fully Eulerian scheme with ${\mathrm{\Phi }}_m=0.0$ (dashed curves) and with the one-way coupling Lagrangian scheme of Ref. [39] (symbols). Lagrangian data have a poorer statistics with respect to Eulerian ones, therefore to decrease a bit the statistical fluctuations we exploited the symmetry with respect to $L/2$ to further average the density profile.

Figure 9

Effect of the order of the dissipative regularization ($h=1$ and $h=2$) on the profile of the mean longitudinal velocity: (a) and (b) ${\overline{u}}_x\left(z\right)$ vs $z$ for ${\mathrm{\Phi }}_m=1$ with $\tau =0.10$ and 0.58, respectively; (c) Amplitude of the mean flow $U$ as a function of ${\mathrm{\Phi }}_m$ for fixed $\tau =0.34$; (c) same of (b) as a function of $\tau$ for fixed ${\mathrm{\Phi }}_m=1.0$. Parameters ${\nu }_p=\kappa ={10}^{-6}$ for $h=2$, for simulations with $h=1$ see Table 1.

Figure 10

Effect of the order of the dissipative regularization ($h=1$ and $h=2$) on the profile of the Reynolds stress $\overline{u_x{u}_z}\left(z\right)$: (a) and (b) $\overline{u_x{u}_z}\left(z\right)$ vs $z$ for ${\mathrm{\Phi }}_m=1$ with $\tau =0.10$ and 0.58, respectively; (c) Amplitude of the Reynolds stress $S$ as a function of ${\mathrm{\Phi }}_m$ for fixed $\tau =0.34$; (c) same of (b) as a function of $\tau$ for fixed ${\mathrm{\Phi }}_m=1.0$. Parameters ${\nu }_p=\kappa ={10}^{-6}$ for $h=2$, for simulations with $h=1$ see Table 1.