Inhomogeneous growth of fluctuations of concentration of inertial particles in channel turbulence

We study the growth of concentration fluctuations of weakly inertial particles in the turbulent channel flow starting with a smooth initial distribution. The steady-state concentration is singular and multifractal so the growth describes the increasingly rugged structure of the distribution. We demonstrate that inhomogeneity influences the growth of concentration fluctuations profoundly. For homogeneous turbulence the growth is exponential and is fully determined by Kolmogorov scale eddies.We derive lognormality of the statistics in this case. The growth exponents of the moments are proportional to the sum of Lyapunov exponents, which is quadratic in the small inertia of the particles. In contrast, for inhomogeneous turbulence the growth is linear in inertia. It involves correlations of inertial range and viscous scale eddies that turn the growth into a stretched exponential law with exponent three halves. We demonstrate using direct numerical simulations that the resulting growth rate can differ by orders of magnitude over channel height. This strong variation might have relevance in the planetary boundary layer.

Figure 1

(a) Components of the Reynolds stress tensor and turbulent kinetic energy $k$ normalized by $u_{\tau }^2$ plotted along nondimensional channel height. The turbulent kinetic energy (red solid line) peaks at $y^{+}=10\text{–}11$. The square of the wall-normal velocity fluctuations $\langle u_y^{\prime }{u}_y^{\prime }\rangle /u_{\tau }^2$ (red dashed line) has its maximum at $y^{+}=100$. (b) Variation of the the Kolmogorov timescale ${\tau }_{\eta }$ (solid line) and the inhomogeneity time (the time at which the particles enters flow region with significantly different statistics) $t_{\mathrm{in}}=y/{\langle u_y^2\rangle }^{1/2}$ (dashed line) along the channel height.

Figure 2

Height-dependent Stokes number for two representative values of particles' relaxation time $\tau$. The averaging of the Stokes number here is over the range of displayed heights.

Figure 3

Comparison of the instantaneous particle distribution of tracer particles (${\text{St}}^{+}=0,$ plot a) and inertial particles (${\text{St}}^{+}=0.1,$ plot b) at time $t^{+}=2.5\times {10}^4$ in the near-wall region between $y+=0\text{–}15$ (dimensions are not to scale). The particles are plotted on top of the time-averaged ${\partial }_y^2\langle u_y^2\rangle$ field.

Figure 4

Particle concentration $n$ normalized by the initial, random, particle concentration $n_0$ vs the wall-normal distance $y^{+}$. Different symbols indicate the temporal evolution from $t^{+}=0$ to $t^{+}=2.3\times {10}^4$. The region below $y^{+}=10$ shows an increase in particle concentration, whereas the particle concentration drops below the initial concentration between $y^{+}=10$ and $y^{+}=60$.

Figure 5

The three integrals of Eq. (33) are plotted vs time normalized by the local Kolmogorov time for all wall distances used for the computation. Darker curves present the regions in the center of the channel and lighter curves are regions near the wall.

Figure 6

(Left) All four terms of the RHS of Eq. (33) vs channel height for the space-averaged Stokes numbers of 0.1 (a) and 0.01 (c): $-{\tau }^2{\int }_0^T\langle {\nabla }^{2}p\left(0\right){\nabla }^{2}p\left(t\right)\rangle dt$, circles; $-\tau {\nabla }_y^2\langle u_y^2\rangle$, squares; $\tau {\nabla }_y{\int }_0^T\langle u_y\left(0\right){\nabla }^{2}p\left(t\right)\rangle dt$, triangles; and ${\tau }^2\left({\partial }_y\langle u_y^2\rangle \right){\partial }_y{\int }_0^t\langle {\nabla }^{2}p\left(t^{\prime }\right)\rangle d{t}^{\prime }$, black points. (Right) We show $\langle \sum {\lambda }_i\rangle$ separately for the homogeneous (circles) and the inhomogeneous contribution (squares), consisting of the sum of $-\tau {\nabla }_y^2\langle u_y^2\rangle$ and $\tau {\nabla }_y{\int }_0^T\langle u_y\left(0\right){\nabla }^{2}p\left(t\right)\rangle dt$ but not including the third term of Eq. (33) for the average Stokes numbers of 0.1 (b) and 0.01 (d). The sum of the two components ${\langle \sum {\lambda }_i\rangle }_{\mathrm{tot}}$ is presented as blue diamonds.

Figure 7

Time dependence of the correlation function that describes inhomogeneous velocity-divergence correlations. We observe good agreement with the theoretical prediction given by Eq. (35).