Modulation of interphase, cross-scale momentum transfer of turbulent flows by preferentially concentrated inertial particles

Wavelet multiresolution analysis is extended to describe interphase, cross-scale interactions involving turbulence kinetic energy (TKE) of particle-laden turbulence. Homogeneous isotropic turbulence (HIT) suspended with inertial particles at the Stokes number of unity (critical particles) is analyzed. Direct numerical simulation is performed for decaying HIT coupled via the Stokes drag law in two ways with the dispersed phase. The effects of two-way coupling on spectral TKE transfer are examined. Clustering of the critical particles in thin, filamentlike regions is observed, and mean wavelet statistics are similar to those of the Fourier analysis. However, spatially local wavelet analysis demonstrates the complexities of the preferential concentration in interphase, interscale energy transfer and associated challenges in subgrid-scale (SGS) modeling of particle-laden turbulence. Two-way coupling enhances correlations between local particle concentration and local interphase TKE transfer. However, particle concentration alone does not indicate a definite direction of interphase energy transfer. Rather, particle clusters behave as an energy source or sink with similar probabilities. A similar argument is made for correlations between particle concentration and cross-scale energy transfer. Wavelet statistics conditioned on coarse-grained number density support the same conclusions. In addition, the joint statistics show the qualitative consistency of the SGS Stokes number in describing the two-way interactions, which should be considered in the SGS modeling of two-way coupled particle-laden turbulence.

Figure 1

Instantaneous particle distributions at $\mathrm{\Delta }t/t_{\ell }=0.64$ on an $x-z$ cross section of thickness equal to ${\ell }_k$ for ${\mathrm{St}}_{k,0}=$ (a) 1 and (b) 10. The arrows denote the integral lengthscales $\ell$ (to scale).

Figure 2

Time development of (a) volume-averaged TKE and (b) viscous dissipation rate.

Figure 3

Normalized Stokes numbers estimated using instantaneous volume-averaged statistics.

Figure 4

Dissipation rate and interphase energy exchange due to the particle drag force for ${\mathrm{St}}_{k,0}=$ (a) 1 and (b) 10.

Figure 5

Comparison of (a) the mean wavelet spectra of TKE and (b) the mean spectral energy transfer due to the particle drag force at $\mathrm{\Delta }t/t_{\ell }=0.64$. In (b), the horizontal dotted line denotes the neutral interphase transfer.

Figure 6

Mean cumulative energy fluxes due to convective and pressure transports at (a) $\mathrm{\Delta }t/t_{\ell }=0.16$ and (b) $\mathrm{\Delta }t/t_{\ell }=0.64$. The horizontal dotted lines denote the in-scale transfer.

Figure 7

Contours of scale-dependent PDF of energy transfer due to the particle drag at $\mathrm{\Delta }t/t_{\ell }=0.64$ for (a) ${\mathrm{St}}_{k,0}=1$ and (b) ${\text{St}}_{k,0}=10$. The horizontal dotted lines denote the neutral interphase transfer.

Figure 8

Instantaneous contours of energy transfer due to the particle drag force at $s=1$ for (a) ${\mathrm{St}}_{k,0}=1$ and (b) ${\mathrm{St}}_{k,0}=10$. The SFS energy transfer due to the triadic interactions at $s=2$ for (c) ${\mathrm{St}}_{k,0}=1$ and (d) ${\mathrm{St}}_{k,0}=10$. All figures are obtained on a part of the $x-y$ midplane and at $\mathrm{\Delta }t/t_{\ell }=0.64$. The contours are overlaid with particles.

Figure 9

Flatness of the interphase coupling term at (a) $\mathrm{\Delta }t/t_{\ell }=0.16$ and (b) $\mathrm{\Delta }t/t_{\ell }=0.64$. Flatness factors at the largest scale ($s=8$) are not shown.

Figure 10

Scale-dependent PDF of the cumulative detailed energy fluxes due to the convective and pressure transports for (a) no-particle and (b) ${\mathrm{St}}_{k,0}=1$ at $\mathrm{\Delta }t/t_{\ell }=0.64$. Cutoff scale $n$ is shown for $n=2$ through 7. The horizontal dotted lines denote the in-scale transfer.

Figure 11

Flatness of the cumulative detailed SFS energy fluxes at (a) $\mathrm{\Delta }t/t_{\ell }=0.16$ and (b) $\mathrm{\Delta }t/t_{\ell }=0.64$. Cutoff scale $n$ is shown for $n=2$–7.

Figure 12

Comparisons of (a) the mean detailed cumulative energy fluxes due to the triadic interactions, and (b) the mean spectral energy transfer due to the particle drag force, computed using the Haar and 24-point Coifman bases. Both plots are obtained for ${\mathrm{St}}_{k,0}=1$ and at $\mathrm{\Delta }t/t_{\ell }=0.16$. The horizontal dotted lines denote (a) the in-scale and (b) the neutral interphase transfers, respectively.

Figure 13

Joint PDF of coarse-grained particle-number density and fluctuation of the logarithm of the local wavelet spectrum of kinetic energy for ${\mathrm{St}}_{k,0}=1$. The first three scales are shown at $\mathrm{\Delta }t/t_{\ell }=0$ (first column) and 0.16 (second column). The solid line denotes the conditional mean of coarse-grained particle-number density.

Figure 14

Joint PDF of coarse-grained particle-number density and energy transfer due to the particle drag force for ${\mathrm{St}}_{k,0}=1$. The first three scales are shown at $\mathrm{\Delta }t/t_{\ell }=0.16$. The solid line denotes the conditional mean of coarse-grained particle-number density.

Figure 15

Joint PDF of coarse-grained particle-number density and the cumulative SFS energy flux for ${\mathrm{St}}_{k,0}=1$. Scales $s=2$ (top row), 3 (middle row), and 4 (bottom row) are shown at $\mathrm{\Delta }t/t_{\ell }=0$ (first column) and 0.16 (second column). The solid line denotes the conditional mean of coarse-grained particle-number density.

Figure 16

Joint PDF of energy transfer due to the particle drag force and the cumulative SFS energy flux for ${\mathrm{St}}_{k,0}=1$. Scales $s=2$ (top row), 3 (middle row), and 4 (bottom row) are shown at $\mathrm{\Delta }t/t_{\ell }=0$ and 0.16. Wavelet statistics are computed using the 24-point Coifman wavelet basis.

Figure 17

Joint PDFs of coarse-grained particle-number density and (a) fluctuation of the logarithm of the local wavelet spectrum of kinetic energy (first column), (b) energy transfer due to the particle drag force (second column), and (c) the cumulative SFS energy flux (third column). For ${\mathrm{St}}_{k,0}=10$, the first three scales are shown at $\mathrm{\Delta }t/t_{\ell }=0.64$. The solid line denotes the conditional mean of coarse-grained particle-number density.

Figure 18

Joint PDFs of coarse-grained particle-number density and (a) fluctuation of the logarithm of the local wavelet spectrum of kinetic energy and (b) the SFS energy flux. The joint statistics are obtained for ${\mathrm{St}}_{k,0}=10$ at $\mathrm{\Delta }t/t_{\ell }=0.64$. The solid line denotes the conditional mean of coarse-grained particle-number density.