Small-scale flow topologies, pseudo-turbulence, and impact on filtered drag models in turbulent fluidization

The small-scale flow structures in a moderately dense turbulent (gas-particle) fluidization are investigated using highly resolved simulation data obtained from a kinetic-theory-based two-fluid model. As a main feature, the invariants of traceless and phase-weighted velocity gradient tensors (rate-of-strain and rate-of-rotation) are computed on the gas and solid phase. The invariant-space joint probability density functions indicate a dominant small-scale topology of tubelike vortical-prevalent clusters and sheetlike viscous dissipative slots on the solid phase. The gas phase, nevertheless, reveals a kind of tendency to a boundary-layer turbulence with a vortex-sheet topology prevalence and balanced mechanism of stretching and contracting enstrophy. From other crucial perspectives, by considering the turbulent interfacial work and drag production [Capecelatro et al., J. Fluid Mech. 780, 578 (2015)], an a priori analysis of the cluster-induced turbulence (pseudo-turbulence) or the subgrid-scale drift velocity parametrizations is outlined. It has been shown that the parametrizations, which depend linearly on the resolved gradient of the solid volume fraction, are invalid in this moderately dense turbulent fluidization. They reveal a clear misalignment with the actual filtered subgrid (solid-induced) drift velocity because of the boundary-layer-like turbulence on gas phase. Alternatively, the tensorial dispersion approach arises to be a valid choice for modeling the drift velocity. The topological analysis, on the other hand, suggests that the linear turbulent dispersion paradigm and the hypothesis of a constant dispersion Prandtl number [Burns et al., ICMF04, (2004), p. 392] is applicable only in the large-scale strain-dominated areas. These and other scrutinies and arguments, from a topological viewpoint, are afforded in this paper.

Figure 1

(a) Schematic representation of the unbounded periodic box (left) and Wall-bounded (right) turbulent fluidizations, rendered on instantaneous ${\epsilon }_s$ outcomes in the TFM simulation. (b) The two-point correlation profiles for the periodic box (bottom) and wall-bounded case (top). The subset in the top of (b) reveals the same $y$-direction profile starting from center towards the walls, while the subset in the bottom of (b) zooms simply the profiles at the starting range.

Figure 2

(a) Classification of local flow topology following $(Q_{\mathsf{G}},R_{\mathsf{G}})$ invariants of velocity gradient tensor for incompressible single-phase flow [taken from the work of Ooi et al. 11]: upper left, stable focus/stretching (SF/S); upper right, unstable focus/compressing (UF/C); lower left, stable node/saddle/saddle (SN/S/S); and lower right, unstable node/saddle/saddle (UN/S/S). (b) Geometric classification of strain in $(Q_{\mathsf{S}},R_{\mathsf{S}})$ invariant space, where the tent lines correspond to two different ratios of principal strains $\left({\lambda }_{{\mathit{s}}_1}\text{:}{\lambda }_{{\mathit{s}}_2}\text{:}{\lambda }_{{\mathit{s}}_3}\right)$, $2\text{:}-1\text{:}-1$, axisymmetric contraction and $1\text{:}1\text{:}-2$, axisymmetric expansion, while the vertical line $1\text{:}0\text{:}-1$ is two-dimensional flow.

Figure 3

Joint PDF maps of normalized invariants on logarithmic scale coloring for spaces $(Q_{\mathsf{G}},R_{\mathsf{G}})$, $(Q_{\mathsf{S}},R_{\mathsf{S}})$, $(Q_{\mathrm{\Omega }},-Q_{\mathsf{S}})$, in respect to the gas phase (a, c, e) and the solid phase (b, d, f), correspondingly. The tent lines therein indicate the null-discriminant curves, $D_{\mathsf{G}}=(27/4)R_{\mathsf{G}}^2+Q_{\mathsf{G}}^3=0$.

Figure 4

Structures of instantaneous high-$Q$ isosurfaces at (b) $Q_{{\mathsf{G}}_s}>800$, (c) $Q_{{\mathsf{G}}_g}>50000$, and (d) renders together $Q_{{\mathsf{G}}_g}>50000$ (light gray) and $Q_{{\mathsf{G}}_g}<-50000$ (dark gray). They are displayed as a highlighted part of the wall-bounded turbulent fluidization (TFM) simulation where (a) represents the synchronous ${\epsilon }_s$ result extracted at the midwidth plane. (See a movie of structures dynamics in Ref. [61]).

Figure 5

Conditional mean vectors of $\langle D{Q}_{{\mathsf{G}}_s}/Dt\rangle$ and $\langle D{R}_{{\mathsf{G}}_s}/Dt\rangle$ in the $(Q_{{\mathsf{G}}_s},R_{{\mathsf{G}}_s}$) space together with some integral trajectories (gray solid lines).

Figure 6

(a) Schematic of vortex line shapes classification in $(q_{\mathit{\omega }},r_{\mathit{\omega }})$ invariant space of (single-phase) vorticity gradient tensor (the scheme is taken from the work of Sharma et al. [65]). (b) Joint PDF map result of $(q_{{\mathit{\omega }}_g},r_{{\mathit{\omega }}_g})$ invariants of gas phase vorticity gradient tensor [Eq. (24)] for the unbounded fully periodic turbulent fluidization case.

Figure 7

PDF of vorticity alignments with the eigenvectors of the rate-of-strain tensor, ${\mathit{s}}_i$, on solid and gas phases for the unbounded fully periodic turbulent fluidization case.

Figure 8

(a) Mean-ensemble ${\mathcal{DE}}_s$ values conditioned on the filtered $(Q_{{\overline{\mathsf{G}}}_s},R_{{\overline{\mathsf{G}}}_s})$ invariants space with a filter length, $\overline{\mathrm{\Delta }}=8\mathrm{\Delta }$ (bottom). At the top of (a), it highlights a midwidth part of $\mathbf{\nabla }·{\mathit{u}}_s$ together with the cluster patterns $({\epsilon }_s>0.4)$. (b) Rendering of structures of $Q_{{\mathsf{G}}_s}>3000$ (left) and $Q_{{\mathsf{G}}_g}>2\times {10}^5$ (right), featured with the middle ${\epsilon }_s$ plane. All data correspond to the unbounded periodic box case.

Figure 9

Mean-ensemble ${\mathcal{DE}}_g$ (a) and $\mathcal{DP}$ (b) conditioned on the filtered $(Q_{{\overline{\mathsf{G}}}_g},R_{{\overline{\mathsf{G}}}_g})$, invariant space, with a filter length, $\overline{\mathrm{\Delta }}=8\mathrm{\Delta }$. All data correspond to the unbounded periodic box case.

Figure 10

JPDF of the angles $({\gamma }_1,{\gamma }_2)$ defined in Eq. (44) and plotted on a unit sphere to exhibit the orientation trends of the SGS drift velocity models ${\mathit{v}}_d^{\mathrm{model}}$ (gray arrow). They show the alignment trends of (a) ${\mathit{v}}_d^{\mathrm{Bur}}$ [Eq. (38)], (c) ${\mathit{v}}_d^{\mathrm{Rau}}$ [Eq. (41)], and (d) ${\mathit{v}}_d^{nl}$ [Eq. (43)], with the actual drift velocity ${\mathit{v}}_d^A$ [Eq. (31)]. A sketch of definitions of the angles is presented in (b). For comparative and simplicity reasons the JPDF events are normalized by their maximals. All data correspond to the unbounded periodic box case.

Figure 11

Alignment trends of the actual SGS drift velocity ${\mathit{v}}_d^A$ (a) and solid gradient vector as ${\mathit{v}}_d^{\mathrm{Bur}}$ closure (b), in the resolved gas phase rate-of-strain eigenframe ${\overline{\mathit{s}}}_i$. The PDF data provided are taken from the unbounded periodic box case.

Figure 12

Instantaneous middle maps for the three $\mathrm{eigenvalues}{\lambda }_{{\mathit{s}}_i}$ of rate of strain ${\mathsf{S}}_g$; the most extensional ${\lambda }_{{\mathit{s}}_1}$ (a), the intermediate ${\lambda }_{{\mathit{s}}_2}$ (b), and the most contractional ${\lambda }_{{\mathit{s}}_3}$ (c), eigenstrains, rendered with the cluster patterns $({\epsilon }_s>0.4)$. All data correspond to the unbounded periodic box case.

Figure 13

Mean ensemble of magnitudes $\parallel {\mathit{v}}_d^A\parallel$ (a) and $\parallel {\mathit{v}}_d^{\mathrm{Bur}}\parallel$ (b), conditioned on the filtered $(Q_{{\overline{\mathrm{\Omega }}}_g},-Q_{{\overline{\mathsf{S}}}_g})$, invariant space, with a filter length, $\overline{\mathrm{\Delta }}=8\mathrm{\Delta }$. The magnitudes are normalized by its maximal, and the small black circle marks the maximum magnitude. All data correspond to the unbounded case.

Figure 14

Mean-ensemble values of cosine angles between $({\mathit{v}}_d^A,{\mathit{v}}_d^{\mathrm{Bur}})$ in (a) and $({\mathit{v}}_d^A,{\mathit{v}}_d^{nl})$ in (b), conditioned on the filtered $(Q_{{\overline{\mathsf{G}}}_g},R_{{\overline{\mathsf{G}}}_g})$ invariants space with a filter length, $\overline{\mathrm{\Delta }}=8\mathrm{\Delta }$. All data correspond to the unbounded periodic box case.

Figure 15

PDF profiles of the angles between the actual SGS drift velocity ${\mathit{v}}_d^A$ and its possible parametrizations based on the Taylor approximation ${\mathit{v}}_d^{\mathrm{nl}}$, and the resolved relative velocity ${\mathit{v}}_r^A$ [Eq. (47)].