Fact-checked by Grok 2 weeks ago

Stokes number

The Stokes number (St), named after the physicist George Gabriel Stokes, is a dimensionless quantity in fluid mechanics that quantifies the extent to which inertial particles suspended in a fluid flow deviate from the fluid's streamlines, defined as the ratio of the particle's relaxation time (τ_p)—the time required for a particle to adjust its velocity to match the surrounding fluid—to a characteristic time scale of the flow (τ_f), such as the time for fluid elements to traverse a relevant length scale. For small spherical particles in low-Reynolds-number flows, the relaxation time is given by τ_p = (ρ_p d_p²) / (18 μ_f), where ρ_p is the particle density, d_p is the particle diameter, and μ_f is the fluid dynamic viscosity, leading to St = τ_p / τ_f = (ρ_p d_p² u_f) / (18 μ_f L) or equivalently St = (Re_L / 18) × (ρ_p / ρ_f) × (d_p / L)², with Re_L as the Reynolds number based on flow velocity u_f, fluid density ρ_f, and length scale L. This parameter is fundamental in analyzing particle-laden flows, where particles interact with turbulent or laminar fluid motions; when St ≪ 1, particles closely follow fluid trajectories due to dominant viscous drag, behaving nearly as passive tracers, whereas St ≫ 1 indicates inertia-dominated motion, causing particles to separate from streamlines, in low-vorticity regions, or surfaces. The Stokes number also influences particle dispersion, , and collision dynamics, with effective St often averaged over instantaneous values in unsteady flows to capture time-dependent behavior. Applications of the Stokes number span diverse fields, including in atmospheric and flows, where it predicts deposition in bends or pipes (e.g., low St favors following curved paths, high St promotes inertial impaction); and droplet formation in geophysical contexts; and particle interactions in processes like spray or pharmaceutical devices. In turbulent boundary layers, St based on viscous or large-eddy timescales helps model preferential concentration and transfer, with values around 1 marking transitions in particle-fluid .

Fundamentals

Definition

The Stokes number, denoted as St, is a dimensionless that quantifies the of the particle relaxation time \tau_p to the characteristic time scale of the flow \tau_f, expressed mathematically as St = \frac{\tau_p}{\tau_f}. This formulation arises in the study of fluid-particle interactions, where \tau_p represents the time required for a particle to adjust its in response to changes in the surrounding motion, and \tau_f captures the temporal scale over which the flow varies significantly. As a , the Stokes number provides a scale-independent measure of how closely particles follow the 's motion; low values indicate particles that rapidly equilibrate with the flow, acting nearly as passive tracers, while high values signify particles with substantial that deviate markedly from fluid streamlines. It plays a central role in characterizing the dynamic response of dispersed particles in multiphase flows, enabling predictions of phenomena such as particle , clustering, and deposition without reliance on for each specific system. The Stokes number is named after the Irish mathematician and physicist George Gabriel Stokes (1819–1903), in recognition of his foundational work on low flows and the settling of particles under viscous drag. This naming reflects its origins in the theoretical framework Stokes established for analyzing the motion of small spheres in quiescent fluids, as detailed in his 1851 paper on the internal friction of fluids.

Physical Interpretation

The Stokes number quantifies the extent to which particles in a respond to changes in the surrounding field, serving as a key parameter in understanding particle dynamics in multiphase systems. Defined as the of the particle relaxation time to the characteristic time scale, it delineates regimes where particle behavior transitions from tight coupling with the to significant inertial independence. When the Stokes number is much less than unity (St ≪ 1), particles possess negligible relative to the flow timescales, closely following fluid streamlines and behaving as passive tracers. In this regime, the equilibrium approximation holds, allowing particles to instantaneously adjust to local fluid velocities without notable lag, which simplifies modeling by treating particles as extensions of the fluid phase. Conversely, at intermediate values (St ≈ 1), particles experience partial decoupling from the flow, exhibiting transitional behavior where causes deviations such as preferential concentration in regions of high and low , leading to enhanced clustering and altered interaction dynamics. For Stokes numbers much greater than unity (St ≫ 1), particles respond minimally to rapid fluctuations, following nearly ballistic trajectories dominated by their initial and external forces like , with limited influence from the surrounding . This inertial regime renders particles akin to inert projectiles, decoupling them from small-scale turbulent motions and promoting or independent of local eddies. Conceptually, low-St particles mimic ideal tracers for accurate field sampling, whereas high-St particles act as independent entities, highlighting the Stokes number's role in distinguishing tracer-like from projectile-like motion in dispersed systems. These behavioral regimes profoundly influence modeling strategies. Low-St conditions favor Eulerian approaches, treating particles as a intertwined with the fluid phase for efficient simulation of dense suspensions. In contrast, higher St necessitates methods to track individual particle trajectories, capturing inertial effects and two-way coupling in dilute, inertial-dominant flows.

Derivation and Calculation

Particle Relaxation Time

The particle relaxation time, \tau_p, represents the characteristic time scale over which a particle adjusts its in response to changes in the surrounding flow, primarily through viscous . It is derived from the particle's momentum equation, which balances the inertial force m_p \frac{du_p}{dt} with the force and other external forces acting on the particle, such as . For small particles where dominates and other forces can be considered separately, the equation simplifies to \frac{du_p}{dt} = \frac{u_f - u_p}{\tau_p}, where u_p is the , u_f is the , and the relaxation time \tau_p emerges as the ratio of particle mass to the . This derivation assumes spherical, rigid particles suspended in a Newtonian, incompressible fluid with no-slip conditions at the particle surface, under creeping flow conditions where the particle \mathrm{Re}_p < 0.1. The flow is steady-state in an infinite medium, with particles moving independently without influencing the carrier fluid or each other. Under these conditions and using Stokes' drag law, F_d = 3\pi \mu d_p (u_f - u_p), the relaxation time for a spherical particle is given by \tau_p = \frac{m_p}{3\pi \mu d_p}, where m_p = \frac{\pi}{6} \rho_p d_p^3 is the particle mass, \mu is the fluid dynamic viscosity, and d_p is the particle diameter. Substituting the mass yields the standard form \tau_p = \frac{\rho_p d_p^2}{18 \mu}, which highlights the dependence on particle density \rho_p and diameter d_p (both increasing \tau_p) and fluid viscosity \mu (decreasing \tau_p). The units of \tau_p are seconds, reflecting its role as a time constant. Typical values range from microseconds for fine aerosols to seconds for larger particles; for example, in air at standard conditions, a 1 \mum diameter particle with density 1000 kg/m³ has \tau_p \approx 3.6 \times 10^{-6} s, while a 10 \mum particle under similar conditions has \tau_p \approx 3.1 \times 10^{-4} s. This timescale serves as the numerator in the Stokes number, \mathrm{St} = \tau_p / \tau_f, quantifying particle inertia relative to flow dynamics.

Characteristic Flow Time Scale

The characteristic flow time scale, denoted \tau_f, is defined as the duration over which the fluid velocity experiences a significant change, providing a reference for the fluid's dynamic behavior in the context of particle motion. In steady, uniform flows, \tau_f is commonly expressed as the ratio of a characteristic length scale L (such as a pipe diameter or obstacle size) to the mean flow velocity U, giving \tau_f = L / U. This formulation captures the time for fluid elements to traverse key geometric features, ensuring the Stokes number reflects spatial velocity gradients appropriately. In turbulent flows, the choice of \tau_f often targets specific eddy timescales to match the scales influencing particle dispersion. For instance, at the dissipative subrange, the Kolmogorov time scale \tau_\eta = (\nu / \epsilon)^{1/2} is employed, where \nu is the fluid kinematic viscosity and \epsilon is the turbulent kinetic energy dissipation rate per unit mass; this scale represents the turnover time of the smallest eddies where viscosity dominates. In oscillatory or periodic unsteady flows, \tau_f is typically the oscillation period T, allowing the Stokes number to quantify particle response to temporal velocity variations. For confined geometries, such as ducts or sampling devices, the residence time—computed as the enclosure volume divided by the volumetric flow rate—serves as \tau_f, highlighting particle lag relative to flow transit through bounded regions. The selection of \tau_f must be tailored to the flow scenario to ensure the Stokes number St = \tau_p / \tau_f (with \tau_p the particle relaxation time) effectively characterizes inertia effects without distortion from mismatched scales. This context-specific approach underscores the importance of aligning \tau_f with dominant unsteadiness or gradients, such as small-scale turbulence or rapid oscillations. By forming the denominator in the Stokes number ratio, \tau_f contributes to its non-dimensional nature, rendering St independent of absolute units and enabling universal comparisons of particle-flow interactions across diverse conditions.

Drag Regimes

Stokesian Drag Regime

The Stokesian drag regime describes the conditions under which the aerodynamic drag on a spherical particle in a fluid can be accurately modeled using the linear , applicable to low-Reynolds-number flows. This regime is fundamental to calculating the particle relaxation time \tau_p without additional corrections, enabling straightforward determination of the St = \tau_p / \tau_f, where \tau_f is the characteristic flow time scale. The drag force F_d in this regime is expressed as F_d = 3 \pi \mu d_p (u_f - u_p), where \mu is the dynamic viscosity of the fluid, d_p is the particle diameter, and u_f - u_p is the relative velocity between the fluid and particle. This formulation assumes creeping flow and is valid when the particle Reynolds number Re_p = \rho_f d_p |u_f - u_p| / \mu \ll 1, typically Re_p < 1, ensuring inertial effects are negligible and the drag is proportional to the relative velocity. The Stokesian regime is characterized by small particle sizes, generally d_p < 10–$100~\mum depending on the fluid and velocity, combined with low relative velocities to maintain low Re_p. Additionally, it requires continuum flow assumptions, where the Knudsen number Kn < 0.01, meaning the particle size is much larger than the fluid molecular mean free path, avoiding rarefaction effects. These conditions are commonly met in dilute suspensions of fine particles, such as aerosols or suspensions in air or water, where particle-particle interactions are minimal. Under these assumptions, the particle relaxation time \tau_p follows directly from balancing the drag force with the particle's inertial response, \tau_p = \rho_p d_p^2 / (18 \mu), without needing empirical corrections for nonlinear drag or slip effects; this simplicity makes the regime ideal for analyzing particle behavior in dilute, low-speed flows. The primary limitation of the Stokesian regime arises at higher Re_p, where nonlinear inertial effects increase the actual drag force beyond the linear prediction, causing the Stokes model to underestimate drag and thereby overestimate \tau_p and St. This breakdown necessitates drag corrections for accurate predictions in more energetic flows.

Non-Stokesian Drag Regime

When the particle Reynolds number, defined as \mathrm{Re_p} = \frac{\rho_f d_p |u_f - u_p|}{\mu}, exceeds approximately 1, the flow around the particle develops significant inertial effects, rendering the linear Stokes drag law invalid. In this non-Stokesian regime, the drag force transitions to a nonlinear form dominated by inertial contributions: F_d = C_d \frac{\pi d_p^2}{4} \frac{\rho_f |u_f - u_p|^2}{2}, where C_d is the dimensionless drag coefficient that varies with \mathrm{Re_p}. This quadratic dependence arises because the boundary layer separation and wake formation behind the particle introduce form drag in addition to viscous drag. The non-Stokesian regime encompasses the intermediate range where $1 < \mathrm{Re_p} < 1000, characterized by gradually increasing flow separation, and the fully turbulent or Newton's regime at \mathrm{Re_p} > 1000, where C_d plateaus at a nearly constant value of about 0.42–0.47 for smooth spheres due to a stable wake structure. Empirical correlations are essential for C_d in the intermediate regime, as analytical solutions are infeasible; a seminal example is the Schiller-Naumann relation: C_d = \frac{24}{\mathrm{Re_p}} (1 + 0.15 \mathrm{Re_p}^{0.687}), valid up to \mathrm{Re_p} \approx 800 and derived from experimental settling velocity data for spheres. To incorporate these effects into the particle relaxation time \tau_p, which is central to the Stokes number \mathrm{St} = \tau_p / \tau_f, the base Stokes expression is adjusted using the drag coefficient: \tau_p = \frac{\rho_p d_p^2}{18 \mu} \cdot \frac{24}{C_d \mathrm{Re_p}}. Substituting the Schiller-Naumann C_d yields \tau_p < \frac{\rho_p d_p^2}{18 \mu} (the uncorrected Stokes value), as the factor $24/(C_d \mathrm{Re_p}) < 1. This decrease in effective \tau_p implies a lower \mathrm{St} for given particle properties and flow scales relative to the uncorrected prediction, reducing the apparent particle inertia. Consequently, predictions using the uncorrected Stokes model overestimate particle inertia and deviation from fluid streamlines, while the correction accounts for enhanced drag, often resulting in less deviation, reduced clustering in turbulent flows, and adjusted sedimentation rates than uncorrected assumptions. Note that \mathrm{Re_p} depends on the slip velocity |u_f - u_p|, requiring iterative solution for precise \tau_p in applications.

Applications

Particle Image Velocimetry

In particle image velocimetry (PIV), the (St) serves as a key metric to evaluate the fidelity with which seeding particles track the fluid velocity field, thereby quantifying potential tracking errors due to particle inertia. For optimal performance, ideal tracer particles should have St < 0.1, which limits velocity slip errors to below 5%, ensuring that measured particle displacements closely represent fluid motion. This threshold is particularly critical in turbulent or unsteady flows, where deviations can introduce systematic biases in velocity estimates. The particle relaxation time (τ_p), a core component of St, is determined by the properties of the seeding particles commonly used in PIV, such as 1-5 μm diameter polystyrene spheres in aqueous flows. For 1 μm polystyrene particles in water, τ_p is approximately 6 × 10^{-8} s, reflecting their near-instantaneous response to fluid forces under Stokesian drag conditions. In PIV setups, τ_p is matched to the flow's characteristic time scale (τ_f) to achieve low St, with particle size selected accordingly—smaller diameters reduce τ_p and thus St for faster flows. Particle tracking errors in PIV arise primarily from velocity slip, approximated as Δu ≈ St × (du_f/dt) × τ_f, where du_f/dt is the fluid acceleration and τ_f is the flow time scale. This slip becomes negligible when St is small, but increases with higher inertia, leading to underestimation of flow gradients. Guidelines for particle selection emphasize using sub-micron sizes in high-speed flows (e.g., supersonic or turbulent jets) to keep St below 0.1, as larger particles exhibit greater lag and amplify measurement uncertainty. Experimental implementation in PIV requires careful consideration of particle-fluid interactions to minimize non-flow effects. Density matching between particles (e.g., polystyrene with ρ_p ≈ 1050 kg/m³) and the medium (water, ρ_f ≈ 1000 kg/m³) is essential to reduce gravitational settling and buoyancy-induced slip, ensuring uniform particle distribution over the measurement volume. Typical St values in air and water PIV experiments range from 0.01 to 0.5, with values near 0.1 common for moderately unsteady flows like indoor air currents or low-Reynolds liquid jets, balancing tracking accuracy with practical seeding density.

Shock Wave Interactions

In shock wave interactions, the Stokes number governs the decoupling between particles and the surrounding gas flow during rapid compressions. For low Stokes number particles (St << 1), the short particle relaxation time allows them to closely follow the abrupt jump across the shock front, maintaining tight coupling with the gas and minimal dispersion. In contrast, high Stokes number particles (St > 1) exhibit significant , causing them to lag behind the , which results in the formation of dispersed particle clouds with uneven profiles—faster particles at the cloud's and slower ones at the trailing edge. The characteristic flow time scale τ_f in these interactions is typically the shock rise time, ranging from nanoseconds to microseconds for strong shocks in gases, or equivalently the downstream relaxation length divided by the shock propagation speed, which can extend to tens or hundreds of microseconds depending on Mach number and particle properties. Experimental and numerical studies demonstrate St-dependent penetration behaviors: for St ≈ 0.2–3.6, particles show varying degrees of lag, with high St cases leading to penetration depths influenced by cloud concentration, where denser clouds attenuate the shock more effectively. In particular, simulations of shock propagation through particle arrays reveal that St > 1 particles can produce separations in the cloud on the order of 10–100 times the initial inter-particle spacing due to differential acceleration. These dynamics have critical applications in blast mitigation and modeling. In multiphase mixtures, high St particles can suppress waves by absorbing energy and attenuating intensity, enhancing safety in systems like hybrid rocket engines. Similarly, particle loading effects in attenuation are leveraged for protective structures, where low St particles provide less mitigation but enable finer control in processes.

Anisokinetic Particle Sampling

Anisokinetic particle sampling arises when the velocity of the sampling differs from the free-stream velocity, resulting in an aspiration efficiency that varies with the Stokes number. This mismatch alters the streamlines entering the , leading to biased particle collection where the sampled concentration deviates from the ambient one. The Stokes number, defined as St = τ_p / τ_f with τ_f as the characteristic time scale from the , quantifies the particle's inertial response relative to changes around the . Particles with high Stokes numbers exhibit ballistic trajectories due to their , causing under-sampling in sub-isokinetic conditions (probe < free-stream) or over-sampling in super-isokinetic conditions, as they fail to adjust to the distorted flow field near the probe entrance. In contrast, low Stokes number particles closely follow the gas streamlines, entering the probe more proportionally to the flow rate and yielding aspiration efficiencies closer to unity. This mechanism highlights the Stokes number's role in predicting sampling biases, particularly for inertial particles in low-speed flows. Quantitative models, such as the semi-empirical correlation by Belyaev and Levin for thin-walled , describe the aspiration ratio η as approximately 1 for St < 0.1, where inertial effects are negligible; however, deviations increase significantly for St > 1, reaching up to 50% or more depending on the ratio R_u (free-stream to ) and . These efficiencies are sensitive to , with blunt showing greater biases at Stokes numbers compared to streamlined designs that minimize distortion. For example, the Belyaev-Levin model gives η = G / (G + √St), where G = (1 + R_u^{2/3}) / (2 R_u^{2/3}), valid for 0.005 < St < 10 and 0.2 < R_u < 5. In applications like aerosol monitoring for air quality assessment and respiratory deposition models simulating particle inhalation, anisokinetic effects must be corrected to ensure representative sampling. Corrections often employ the Stokes number St = τ_p U / D, where U is the free-stream velocity and D the probe diameter, to adjust measured concentrations for inertial losses and enable accurate health risk evaluations.

Turbulence Modulation in Particle-Laden Flows

In particle-laden turbulent flows, two-way coupling becomes significant when the particle volume fraction exceeds approximately $10^{-6}, allowing inertial particles to influence the carrier fluid's turbulence through momentum and energy exchange. Particles with intermediate Stokes numbers (St ≈ 0.1–10) preferentially attenuate small-scale turbulence by extracting kinetic energy from the fluid phase, leading to reduced turbulent kinetic energy (TKE) levels. This attenuation is particularly pronounced for heavy particles where the particle-to-fluid density ratio \rho_p / \rho_f \gg 1, as the added mass enhances the feedback effect on the flow. The primary mechanisms driving this modulation include enhanced dissipation due to viscous wakes around particles and preferential concentration, where particles with St ≈ 1 cluster in low-vorticity regions of the flow, thereby altering the local energy dissipation rates. For instance, direct numerical simulations (DNS) have shown that such clustering disrupts small eddies, contributing to a non-monotonic dependence on St: attenuation dominates at intermediate St, while very high St particles may induce augmentation through wake instabilities. Quantitative studies indicate turbulence intensity reductions of up to 50% in TKE for St in the range 1–10 and mass loadings \Phi_m > 1, with the effect scaling with particle inertia relative to eddy timescales. Modeling these interactions often incorporates the Stokes number into (PDF) methods or (LES) to capture subgrid-scale particle-fluid coupling, with two-way coupling terms accounting for back-reaction on the fluid solver. The density ratio \rho_p / \rho_f \gg 1 amplifies modulation by increasing the relative , as validated in four-way coupled point-particle DNS. In applications such as pulverized , where high mass loadings prevail, this leads to suppressed flame and altered efficiency; similarly, in environmental phenomena like dust storms, St-dependent clustering influences visibility and . Recent post-2020 DNS studies highlight how varying St modulates preferential concentration in homogeneous isotropic , providing insights into scalable models for dense suspensions.