Turbulent Concentration (TC) in Turbulent Flows

• TC is the phenomenon where inertial particles cluster in turbulence by preferentially accumulating in regions of high strain rate and low vorticity.
• Mathematical models, including Eulerian-Eulerian and Eulerian-Lagrangian frameworks, capture the balance between inertial compressibility and diffusive smoothing.
• TC impacts key processes such as collision enhancement, rain initiation, and planet formation through mechanisms like biased sampling, turbophoresis, and centrifugal forces.

Turbulent concentration (TC), also called preferential concentration in much of the inertial-particle literature, is the tendency of finite-inertia particles to develop nonuniform spatial distributions in turbulence by accumulating in regions of high strain rate and low vorticity rather than following the carrier flow as passive tracers. In the supplied studies, TC appears as a particle-clustering mechanism in two-way coupled particle-driven convection, in mechanically driven turbulence, in wall-bounded rotating turbulence, and in particle-laden protoplanetary-disk midplane layers. Across these settings, the recurring physical picture is that particle inertia makes the particle phase effectively compressible, while diffusion or dispersive processes oppose singular concentration growth; the resulting balance generates intermittent clusters, filaments, and voids with direct implications for collision enhancement, rain initiation, and particle growth (Nasab et al., 2020, Nasab et al., 2021, Umurhan et al., 7 Aug 2025).

1. Physical basis of preferential concentration

The defining statement in the supplied literature is that heavy inertial particles accumulate in regions of high strain rate and low vorticity due to their inertia (Nasab et al., 2020). The mechanism is kinematic but not tracer-like: particles respond to the carrier flow only over a finite stopping or relaxation time, so coherent vortical structures centrifuge them outward while strain-dominated regions collect them. In this sense, TC is not merely a fluctuation in particle number density; it is a systematic bias in how inertial particles sample the flow.

In the Eulerian-Eulerian two-fluid formulation, the same mechanism is expressed through the particle velocity field rather than through individual trajectories. For small $\tau_p$, the particle velocity can be written schematically as

$$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$

so that

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$

even though the carrier flow is incompressible, $\nabla\cdot \mathbf{u}=0$. The particle phase is therefore weakly compressible, and TC emerges from the competition between this inertial compressibility and particle diffusion $\kappa_p$ (Nasab et al., 2021).

The same phenomenology reappears in disk simulations in a more visibly geometric form. There, short particle filaments collect along regions of high gas strain rate and enclose gas-only voids exhibiting coherent vorticity. The studies define gas vorticity by $$\boldsymbol{\omega}\equiv \nabla\times \boldsymbol{U}$$ and the strain-rate tensor by

$$S_{ij}=\frac{1}{2}\left(\frac{\partial u_{g,j}}{\partial x_i}+\frac{\partial u_{g,i}}{\partial x_j}\right),$$

with the scalar strain rate taken from $S_{ij}$. In that formulation, TC is diagnosed directly by the anti-correlation between particle abundance and void vorticity, and the positive correlation between particle abundance and high strain (Umurhan et al., 7 Aug 2025).

2. Mathematical descriptions and regime of validity

Two complementary modeling frameworks dominate the supplied corpus. One is the Eulerian-Eulerian two-fluid approach, in which the particles are treated as a continuum field with its own momentum and mass conservation laws. The other is the Eulerian-Lagrangian point-particle approach, in which the carrier flow is solved in Eulerian form while particles are tracked individually with inertial equations of motion.

The two-fluid studies focus on small Stokes number regimes where the continuum approximation remains valid. One paper states that the model is appropriate when the Stokes number is ${\rm St}\lesssim O(0.1)$ and anticipates breakdown as ${\rm St}\to 1$ (Nasab et al., 2020). A second mechanically driven study restricts attention to $$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$0 in moderately turbulent flows with $$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$1, again emphasizing that the continuum treatment is valid only when particles remain sufficiently correlated with the fluid (Nasab et al., 2021). In both cases, the particle density equation includes an effective diffusivity $$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$2, interpreted as a phenomenological representation of Brownian motion and unresolved particle interactions.

The Taylor–Couette realization uses a one-way coupled Eulerian-Lagrangian DNS. There the carrier flow obeys

$$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$3

with $$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$4, radius ratio $$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$5, aspect ratio $$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$6, and particle Stokes numbers

$$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$7

The particle equations include Stokes drag and curvature terms in cylindrical coordinates, and the low particle volume fraction $$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$8 justifies one-way coupling and neglect of collisions (Jiang et al., 2024).

A central formal result in the Taylor–Couette case is the radial concentration balance

$$\mathbf{u}_p \approx \mathbf{u} - \tau_p\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} - \cdots\right),$$9

where

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$0

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$1

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$2

These terms represent biased sampling, turbophoresis, and centrifugal effect, respectively, and provide a flow-specific decomposition of TC-related radial migration in a rotating wall-bounded system (Jiang et al., 2024).

3. Scaling laws and statistical structure

The strongest quantitative regularity in the supplied literature is the repeated appearance of the parameter

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$3

which controls multiple aspects of TC. In the particle-induced convective instability, for Stokes number above $$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$4, the maximum particle concentration enhancement over the mean scales as

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$5

the typical or rms enhancement scales as

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$6

and the exponential tail slope of the concentration PDF scales as

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$7

when the tail is fit by $$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$8 (Nasab et al., 2020).

The mechanically driven study verifies that the same scaling laws apply when turbulence is externally driven rather than particle-driven. In its nondimensional notation,

$$\nabla \cdot \mathbf{u}_p \sim -\tau_p\, \nabla \cdot (\mathbf{u}\cdot\nabla \mathbf{u}),$$9

and the high-concentration tail obeys

$\nabla\cdot \mathbf{u}=0$0

This extends the earlier dominant-balance picture from a specialized convective instability to mechanically forced turbulence and supports the view that TC is controlled by inertial compression versus diffusive smoothing rather than by the particular turbulence source (Nasab et al., 2021).

The statistical interpretation is equally consistent across studies. Maximum enhancement is associated with rare intense clusters, whereas rms enhancement measures the typical fluctuation level. The exponential PDF tail implies that strong clustering events are intermittent but systematically organized. This suggests that rare events, rather than mean concentration alone, can control physically important outcomes such as collision enhancement.

A separate but related scale-selection result appears in the disk-layer study. There the effective vorticity-based Stokes number is defined as

$\nabla\cdot \mathbf{u}=0$1

while the standard disk Stokes number is $\nabla\cdot \mathbf{u}=0$2. The PDFs of $\nabla\cdot \mathbf{u}=0$3 drawn from the void set peak at about

$\nabla\cdot \mathbf{u}=0$4

matching the previously identified clustering-optimal value $\nabla\cdot \mathbf{u}=0$5 for scale-dependent TC intermittency (Umurhan et al., 7 Aug 2025).

4. Turbulent concentration in Taylor–Couette turbulence

In turbulent Taylor–Couette flow, TC is shaped by wall-bounded turbulence, global rotation, and Taylor-roll organization rather than by homogeneous isotropic turbulence alone. The supplied DNS study examines inertial particles in a flow between concentric cylinders with the inner cylinder rotating and the outer cylinder fixed, and finds that particles cluster more strongly as $\nabla\cdot \mathbf{u}=0$6 increases, particularly near the outer wall and in regions of low fluid velocity (Jiang et al., 2024).

The preferential-concentration diagnostic is based on two-dimensional Voronoï analysis in a thin shell near the outer wall,

$\nabla\cdot \mathbf{u}=0$7

For randomly distributed particles in 2D, the normalized Voronoï area $\nabla\cdot \mathbf{u}=0$8 follows the stated $\nabla\cdot \mathbf{u}=0$9-distribution, and deviations from that reference quantify clustering and void formation. As $\kappa_p$0 increases, the distributions depart more strongly from random, clusters become more pronounced, and the Voronoï cells indicate anisotropic, azimuthally elongated structures. These clusters align with axial stripe-like structures associated with Taylor rolls and preferentially occupy low-speed streaks (Jiang et al., 2024).

The radial particle distribution is not attributed to preferential concentration alone. The concentration profile shows decrease near the inner wall with increasing $\kappa_p$1, depletion in the bulk, and strong accumulation near the outer wall. The balance-model decomposition shows that centrifugal force is the dominant mechanism across all cases. Biased sampling pushes particles toward the inner wall because particles preferentially sample inward radial flow in Taylor rolls, while turbophoresis drives particles toward both walls where turbulent intensity is weaker than in the bulk and becomes more important for larger $\kappa_p$2. Across the full radial domain,

$\kappa_p$3

so centrifugal forcing remains strongest (Jiang et al., 2024).

This wall-bounded rotating case therefore refines the general TC picture. It shows that observed concentration fields can result from the superposition of clustering, nonuniform flow sampling, turbulence-intensity gradients, and system-scale rotation. A plausible implication is that in rotating reactors and related devices, “preferential concentration” should not be treated as a single-mechanism explanation for wall accumulation.

5. Geophysical and astrophysical relevance

The supplied studies repeatedly connect TC to particle growth problems. In mechanically driven turbulence, preferential concentration is described as playing a key role in promoting particle growth relevant to warm rain formation in clouds, planet formation, and industrial sprays. Even when $\kappa_p$4, significant clustering may still occur if $\kappa_p$5 is small enough and turbulence is strong enough. The cloud discussion further argues that maximum enhancement can be very large, that the corresponding volume fraction in clustered regions can approach values where particle interactions become important, and that the extreme events lie in the exponential tail of the PDF. Those events are rare, but the proposed implication is that a few “lucky” droplets can seed runaway growth (Nasab et al., 2021).

The particle-driven convective study extends this relevance to geophysical and astrophysical settings in which the particles themselves help drive the turbulence. It explicitly discusses volcanic ash clouds, stratus clouds, turbidity currents, collapsing protostars or dusty gas clouds, and planet formation environments. The stated conclusion is that preferential concentration is likely weak in many geophysical cases because the Stokes number is too small, but near volcanic plumes the non-dimensional stopping time can reach the onset of the inertial regime and the corresponding enhancements may be large enough to matter for aggregation and sedimentation (Nasab et al., 2020).

In protoplanetary-disk midplane layers, TC is presented as a persistent small-scale clustering mechanism that can operate even when the large-scale axisymmetric filaments usually associated with the streaming instability either cannot form or have not yet developed. The diagnostic pattern is explicit: particles collect along regions of high gas strain rate and enclose gas-only voids exhibiting coherent vorticity. The study further argues that, for $\kappa_p$6 and midplane particle-to-gas density ratios below unity, streaming-instability growth rates are $\kappa_p$7–$\kappa_p$8 orders of magnitude slower than turbulent overturn frequencies at the driving scale, and therefore cannot plausibly be the main driver of the turbulence in those cases. The authors instead suggest symmetric instability as the likely driver, while TC accounts for the small-scale clumping (Umurhan et al., 7 Aug 2025).

A further reinterpretation follows from that analysis. Roche-exceeding small-scale fluctuations inside large-scale streaming-instability filaments are argued to be not direct expressions of the streaming instability, but manifestations of TC amplified by the elevated particle densities within those larger structures. This suggests that small-scale extreme overdensities in disk simulations may require a multiscale interpretation in which large-scale filament formation and small-scale TC are dynamically distinct even when they coexist.

6. Terminology, misconceptions, and unresolved issues

The abbreviation “TC” is itself context-dependent. In the inertial-particle literature discussed above, it denotes turbulent concentration. In a separate fluid-mechanics literature, however, “TC flow” means Taylor–Couette flow, the flow between two coaxial cylinders rotating independently (Ezeta et al., 2020, Berghout et al., 2020). This overlap is not merely linguistic: one of the supplied studies examines turbulent concentration of inertial particles precisely in turbulent Taylor–Couette flow, so the acronym is overloaded within the same broader field (Jiang et al., 2024).

A recurring misconception is that TC is exhausted by the statement “particles collect in strain and avoid vortices.” The supplied studies support that statement, but they also show that the measurable concentration field can be reorganized by diffusion, turbophoresis, biased sampling, centrifugal migration, and particle feedback. In the Taylor–Couette system, outer-wall accumulation is dominated by centrifugal effects rather than by clustering alone (Jiang et al., 2024). In disk layers, the void–filament structure may be sustained by particle feedback, and the convergence of $\kappa_p$9 toward $$\boldsymbol{\omega}\equiv \nabla\times \boldsymbol{U}$$0 is presented as likely reflecting how particle feedback structures and sustains voids (Umurhan et al., 7 Aug 2025).

Another important clarification concerns generality. The original two-fluid scaling laws were obtained in a particle-driven convective instability, but the mechanically driven study shows that the same laws survive when turbulence is externally forced. This does not eliminate all uncertainty, however. The stated caveats remain substantial: moderate Reynolds numbers only ($$\boldsymbol{\omega}\equiv \nabla\times \boldsymbol{U}$$1 in the mechanically driven DNS), validity of the two-fluid approximation only for sufficiently small Stokes number, phenomenological modeling of particle diffusivity $$\boldsymbol{\omega}\equiv \nabla\times \boldsymbol{U}$$2, omission of gravity in the mechanically driven study, and the unresolved question of whether the relevant velocity scale at very high Reynolds number is the injection-scale $$\boldsymbol{\omega}\equiv \nabla\times \boldsymbol{U}$$3 or a smaller-scale velocity such as one at the Taylor microscale (Nasab et al., 2021, Nasab et al., 2020).

Taken together, these studies define TC as a robust but context-sensitive clustering mechanism. Its core signature—particle accumulation in high-strain, low-vorticity regions—remains stable across formulations and applications, while the observable morphology, radial distribution, and dynamical consequences depend strongly on coupling regime, flow geometry, and the competing transport processes that accompany inertial clustering.