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Secondary flow

Secondary flow in fluid dynamics refers to the relatively weak, transverse component of fluid motion that is superimposed on the primary, streamwise flow direction in three-dimensional flow fields, often resulting from imbalances in pressure gradients, centrifugal forces, or anisotropic turbulence.^[1] This phenomenon typically manifests as circulatory patterns or vortices perpendicular to the main flow, with velocities on the order of 10% of the axial velocity, and plays a critical role in enhancing mixing, altering shear stress distributions, and influencing energy dissipation in various engineering and natural systems.^[2] The primary mechanisms driving secondary flow include transverse pressure gradients in curved conduits, where centrifugal forces push fluid toward the outer wall, creating counter-rotating vortices known as Dean vortices.^[3] In non-circular ducts, such as square or rectangular channels, Prandtl's secondary flows arise from the anisotropy of Reynolds stresses in turbulent boundary layers, directing fluid from the center toward the corners along the walls.^[1] Rotational effects, like Coriolis forces in spinning fluids or system rotation in Taylor-Couette flows, further induce spiral or ribbon-like secondary motions,^[1] while in particle-laden or viscoelastic fluids, additional instabilities can amplify these patterns.^[4] Notable examples of secondary flow occur in pipe bends, where it leads to uneven velocity profiles and increased friction losses, as first theoretically analyzed by Dean in 1928 for steady laminar flow in curved channels.^[3] In open-channel bends, such as rivers, secondary currents form helical cells that transport sediment and shape geomorphology.^[5] Applications span turbomachinery, where secondary flows in blade passages reduce efficiency by promoting endwall losses;^[6] cardiovascular flows in arterial bends, enhancing mass transport;^[7] and atmospheric boundary layers, generating roll vortices that affect pollutant dispersion.^[8] Overall, secondary flows are quantified using dimensionless parameters like the Dean number (De = Re √(a/R), where Re is the Reynolds number, a the pipe radius, and R the curvature radius), which governs their intensity and stability.^[3]

Definition and Fundamentals

Core Definition

In fluid dynamics, secondary flow refers to the component of a three-dimensional flow field that is perpendicular to the primary flow, which is the dominant motion parallel to the main direction of fluid transport.^[1] This perpendicular motion often manifests as cross-stream circulations or vortices superimposed on the stronger primary flow, particularly in non-uniform or curved flow geometries.^[9] The primary flow follows the mainstream direction, such as the axial velocity in a straight pipe or channel, while secondary flow arises as a weaker, transverse component that redistributes momentum across the flow cross-section.^[1] For instance, in pipe flows, the primary flow is streamwise, but secondary flow induces helical or vortical patterns that can enhance mixing or alter shear stresses.^[1] Secondary flows are fundamentally described by the incompressible Navier-Stokes equations, which govern the momentum balance for viscous, low-Mach-number fluids without deriving explicit forms here. These equations capture the interplay of pressure gradients, viscous diffusion, and convective acceleration leading to such transverse motions.^[10] In terms of scales, secondary flow velocities are typically much smaller than primary flow velocities, often reaching only about 10% of the axial speed under laminar conditions.^[9] This disparity, denoted as v_{\text{secondary}} \ll v_{\text{primary}}, underscores the perturbative nature of secondary flows relative to the mainstream.^[1]

Historical Development

The earliest qualitative observations of secondary flow emerged in the late 19th century through Joseph Boussinesq's investigations into fluid motion in curved conduits, where he identified cross-stream circulations arising from the interaction between centrifugal forces and pressure gradients in laminar pipe flows.^[11] Boussinesq's 1868 analysis laid the groundwork by mathematically describing these transverse motions, emphasizing their role in deviating from purely axial flow patterns.^[12] A pivotal advancement occurred in the 1920s with William R. Dean's theoretical work on steady laminar flow through curved pipes of circular cross-section. In his 1927 paper, Dean employed a perturbation expansion based on small curvature ratios to derive analytical solutions, revealing a pair of counter-rotating vortices in the cross-sectional plane as the characteristic secondary flow structure.^[13] Extending this in 1928, Dean further refined the model for stream-line motion, introducing the dimensionless Dean number, defined as De = Re \sqrt{a/R} (where Re is the Reynolds number, a is the pipe radius, and R is the radius of curvature), which quantifies the balance between inertial and centrifugal effects driving the secondary circulation.^[3] These solutions provided the first quantitative framework for predicting secondary flow intensity and stability, influencing subsequent studies on laminar-to-turbulent transitions in curved ducts. By the mid-20th century, attention shifted toward applications in rotating machinery, with Dieter Eckardt's experimental and analytical studies in the 1970s elucidating secondary flow patterns in high-speed centrifugal compressors. Eckardt's 1976 investigation detailed the formation of vortex-like secondary flows within impeller passages, driven by Coriolis and curvature effects, and quantified their impact on flow nonuniformity and efficiency losses through laser anemometry measurements.^[14] This work marked a key milestone in extending secondary flow concepts from simple pipes to complex, rotating geometries in turbomachinery, highlighting their role in generating endwall vortices and radial transport. Post-1980s developments leveraged computational fluid dynamics (CFD) to deepen insights into secondary flows across intricate configurations beyond analytical tractability. Early numerical efforts, as reviewed by Berger et al. in 1983, began simulating fully developed laminar regimes in toroidal pipes, validating Dean's approximations and exploring higher-order instabilities.^[15] Subsequent advancements in the 1990s and beyond, using finite volume and spectral methods, enabled detailed predictions of secondary flow evolution in transitional and turbulent regimes within non-circular ducts and bends, revealing phenomena like vortex pairing and swirl-switching that enhance mixing and heat transfer. These simulations have become indispensable for optimizing designs in engineering contexts, providing high-fidelity visualizations unattainable through experiments alone.^[11]

Physical Mechanisms

Origin in Curved and Rotating Flows

Secondary flow arises fundamentally from geometric and inertial effects in curved and rotating fluid systems, where forces perpendicular to the primary flow direction induce cross-stream circulations. In curved channels, the primary axial flow experiences centrifugal acceleration due to the channel's curvature, generating a radial pressure gradient that drives secondary motions. Near the channel walls, this results in outward-directed flow, while in the core region, fluid moves inward to balance the imbalance, forming characteristic counter-rotating vortices known as Dean vortices. This phenomenon was first analytically described by W.R. Dean in his seminal work on laminar flow in curved pipes, where the centrifugal force acts on fluid elements moving along the curved path, creating a transverse circulation superimposed on the primary flow. The strength of this secondary flow depends on the balance between inertial and viscous forces, quantified by the Dean number De = Re \sqrt{a/R}, where Re is the Reynolds number, a is the pipe radius, and R is the radius of curvature. In laminar regimes, the secondary velocity scales as \sqrt{a/R} times the primary axial velocity, highlighting how even mild curvature can produce significant cross-flow effects when inertia dominates viscosity. In rotating flows, secondary circulations emerge from the Coriolis force, which deflects the primary flow in a rotating reference frame, leading to helical pathlines and transverse velocities. A classic example is the Ekman layer, where over a rotating boundary, the Coriolis effect balances viscous friction, producing a spiral wind or current profile with secondary flow components that transport mass across the primary direction. V.W. Ekman developed this theory to explain wind-driven ocean currents, demonstrating how planetary rotation induces boundary-layer circulations with radial inflows or outflows depending on the hemisphere.^[16] The inertial-viscous balance in these rotating cases similarly governs secondary flow intensity, with the Rossby number Ro = U/(f L) (where f is the Coriolis parameter and L a length scale) indicating when rotation significantly perturbs the flow; low Ro enhances secondary circulations akin to the curvature-induced scaling in non-rotating systems. While these bulk effects initiate secondary flows, boundary layers amplify them through viscous interactions, as detailed in subsequent analyses.

Role of Viscosity and Boundary Layers

Viscosity plays a crucial role in the formation of secondary flows by enabling momentum transfer across streamlines, which amplifies the pressure imbalances generated by centrifugal effects in curved flows. In the absence of viscosity, an inviscid flow would not develop secondary circulation, as there would be no mechanism to enforce the no-slip condition or diffuse momentum radially. Instead, viscosity allows for the resolution of the radial pressure gradient—arising from the imbalance between centrifugal force \rho u^2 / R (where \rho is density, u the primary velocity, and R the radius of curvature) and the resulting transverse pressure differences—through diffusive processes that drive circulatory motion perpendicular to the primary flow direction. This viscous diffusion is particularly prominent near walls, where it sustains the secondary flow against dissipation.^[17] The no-slip condition at solid boundaries induces boundary layer skewing, a process where the curvature of the flow path deflects the shear layers, generating initial secondary velocities confined to a thin region adjacent to the wall. As the flow progresses along the curved path, these transverse components intensify due to the ongoing skewing, and viscous diffusion spreads the momentum outward, allowing the secondary velocities to permeate the entire cross-section and form coherent vortex structures, such as the paired Dean vortices in circular pipes. This growth from boundary-confined to fully developed secondary flow typically occurs over a development length scaling with \sqrt{R d} (where d is the pipe diameter), highlighting the interplay between viscous spreading and geometric curvature. The boundary layer thickness itself, \delta \sim \sqrt{\nu x / u} (with \nu the kinematic viscosity and x the streamwise distance), modulates the rate at which skewing effects propagate, ensuring that secondary flows are more pronounced in high-viscosity fluids or at lower primary velocities. In addition to these viscous mechanisms in curved flows, secondary flows can also arise in straight ducts with non-circular cross-sections due to turbulence-driven effects. Known as Prandtl's secondary flows of the second kind, these are caused by the anisotropy of the Reynolds stresses in turbulent boundary layers, where the turbulent normal stresses differ in streamwise and spanwise directions. This imbalance directs fluid from the center of the duct toward the corners along the walls, forming up to eight counter-rotating vortices in square ducts, with secondary velocities typically 1-3% of the bulk velocity. Unlike curvature-induced flows, these do not require molecular viscosity but depend on the turbulent eddy viscosity and are absent in laminar regimes.^[18] Prandtl's mixing-length theory provides a framework for estimating the magnitude of secondary velocities, particularly in extending the concept to turbulent regimes where an effective eddy viscosity replaces molecular viscosity. A rough scaling from this approach yields v_s \sim \nu / (R u_\mathrm{primary}) for the ratio v_s / u_\mathrm{primary}, indicating that secondary flows weaken relative to the primary flow as Reynolds number increases, since the transverse momentum transport is limited by the viscous scale relative to inertial curvature effects. This estimation arises from balancing the centrifugal driving term with mixing-length-based transverse diffusion, where the mixing length l \sim \sqrt{\nu R / u} approximates the scale over which momentum is exchanged across streamlines. In practice, for turbulent curved flows, substituting eddy viscosity \nu_t \approx 0.4 u d recovers the observed scaling v_s / u \sim d / R, consistent with experimental measurements in mildly curved channels.^[19] Secondary flows induced by these viscous mechanisms can precipitate transition to turbulence at Reynolds numbers lower than in straight channels for certain curvatures, as the transverse circulation introduces shear instabilities and enhances mixing. This effect was first demonstrated experimentally through visualization of streamlines, where even mild curvatures were shown to disrupt streamline motion and induce turbulent eddies at reduced flow rates.^[20]

Mathematical Description

Governing Equations

The governing equations for secondary flows in incompressible fluids are the three-dimensional Navier-Stokes equations coupled with the continuity equation, which describe the conservation of momentum and mass, respectively.^[21] For steady-state conditions, the momentum equation takes the form

(\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u},

where \mathbf{u} is the velocity vector, p is the pressure, \rho is the fluid density, and \nu is the kinematic viscosity, while the continuity equation enforces incompressibility as \nabla \cdot \mathbf{u} = 0.^[21] These equations capture the full dynamics of viscous flows, including the development of secondary circulations driven by pressure gradients and centrifugal forces in non-straight geometries. In configurations involving curvature or rotation, such as flow through curved ducts, the Navier-Stokes equations are transformed to curvilinear coordinates aligned with the geometry to better resolve the transverse variations.^[22] The general form in orthogonal curvilinear coordinates (\xi, \eta, \zeta) involves metric tensors to account for scale factors, resulting in expanded expressions for the convective, pressure, and viscous terms; for instance, the Laplacian operator \nabla^2 \mathbf{u} becomes a combination of second derivatives weighted by the coordinate scales.^[22] This transformation facilitates the analysis of secondary motions perpendicular to the primary flow direction without introducing artificial distortions from Cartesian grids.^[21] The velocity field \mathbf{u} is typically decomposed into a primary component \mathbf{u}_\text{primary}, aligned with the main streamwise direction, and a secondary component \mathbf{u}_\text{secondary}, representing transverse circulations, such that \mathbf{u} = \mathbf{u}_\text{primary} + \mathbf{u}_\text{secondary}.^[21] This separation highlights how secondary velocities, often small in magnitude compared to the primary flow (on the order of \delta times the primary speed, where \delta is a curvature measure), arise from imbalances in the momentum equations and contribute to overall flow redistribution. To assess the relative importance of inertial, viscous, and curvature effects, the equations are non-dimensionalized using characteristic scales for velocity U, length L, and time L/U. This yields the Reynolds number \text{Re} = UL / \nu, which governs the balance between inertia and viscosity, and a curvature parameter \delta = a / R, where a is the duct cross-sectional radius and R is the radius of curvature, quantifying the geometric influence on secondary flow strength. In the non-dimensional form, the momentum equation becomes \partial \hat{\mathbf{u}} / \partial \hat{t} + (\hat{\mathbf{u}} \cdot \nabla) \hat{\mathbf{u}} = -\nabla \hat{p} + (1/\text{Re}) \nabla^2 \hat{\mathbf{u}}, with secondary terms scaled by \delta.^[21] Appropriate boundary conditions are essential for solving these equations. At solid walls, the no-slip condition requires \mathbf{u} = 0, ensuring zero velocity tangent and normal to the surface.^[21] For fully developed flows in periodic or long domains, such as straight sections upstream or downstream of bends, velocity and pressure fields satisfy periodic boundary conditions in the streamwise direction.^[22]

Key Analytical Models

One of the seminal analytical models for secondary flow is Dean's formulation for laminar flow in curved pipes, developed in the late 1920s. In this model, the curvature induces a centrifugal force imbalance that drives a pair of symmetric, counter-rotating vortices in the cross-sectional plane, known as Dean vortices. The circulation strength \Gamma of these bifurcated vortices scales as \Gamma \sim (De / Re) u a, where De is the Dean number (a dimensionless parameter combining the Reynolds number and the curvature ratio a/R, with a as the pipe radius and R as the radius of curvature), u is the mean axial velocity, and a is the pipe radius. This scaling arises from a perturbation expansion of the Navier-Stokes equations assuming small curvature, where the secondary velocity components are of order (De / Re) times the primary flow speed. For circular cross-sections, the model predicts a stable two-vortex pattern, while extensions to square ducts reveal a four-cell structure at higher Dean numbers, with additional vortices forming near the corners due to enhanced centrifugal effects. An analogous analytical framework emerges from Taylor-Couette flow between concentric rotating cylinders, where differential rotation generates a centrifugal instability leading to meridional secondary circulations, or Taylor vortices. In this setup, the inner cylinder's rotation imposes a Coriolis-like force analogous to the centrifugal force in curved pipes, resulting in axisymmetric roll cells that organize the secondary flow. Linear stability analysis of the base azimuthal flow yields a critical Taylor number (similar to the Dean number) beyond which these circulations emerge, with vortex strength proportional to the square root of the supercriticality parameter. This model provides a controlled analogy for understanding secondary flows driven by rotation, highlighting the role of system geometry in vortex pairing and stability. Dean's model and its extensions hold primarily for low Dean numbers (De < 1000), where laminar assumptions remain valid and secondary flows remain weak compared to the primary axial flow. At higher Dean numbers, the flow transitions to turbulence, where the simple perturbation approach breaks down, and vortex structures become unstable or distorted. To address turbulent regimes, Batchelor's 1967 model extends the analysis for straight non-circular ducts by assuming an inviscid turbulent core with anisotropic Reynolds stresses driving the secondary circulation. In this framework, the secondary flow arises from the non-uniform distribution of turbulent momentum transport across the cross-section, producing corner eddies that scale with the bulk Reynolds stress anisotropy, without relying on explicit curvature or rotation. This inviscid core approximation captures the persistence of secondary flows in turbulent boundary layers, providing a bridge to more complex geometries.

Natural Examples

Near-Surface Wind Patterns

In the atmospheric boundary layer near the Earth's surface, secondary flows manifest as deviations from the geostrophic wind due to frictional interactions with the ground, primarily within the Ekman layer. This layer arises from the balance among the Coriolis force, pressure gradient force, and turbulent friction, resulting in a characteristic spiral deflection of the wind vector with height. In the Northern Hemisphere, the surface wind direction veers to the left of the geostrophic wind by approximately 45°, gradually aligning with the geostrophic direction aloft as frictional influence wanes.^[23] This deflection stems from the viscous coupling in rotating flows, where each successive layer drags the one below it, producing a cross-isobaric component that drives secondary circulation.^[24] The Ekman layer typically extends to a depth of about 1 km, beyond which the flow approximates geostrophic balance with negligible friction. Within this layer, friction induces secondary circulations characterized by crosswind velocities that are roughly 10-20% of the primary geostrophic flow speed, facilitating momentum transfer and vertical mixing. These circulations form helical patterns aligned with the mean flow, enhancing turbulence and influencing the overall boundary layer structure.^[25]^[26] Traditional observations using anemometer arrays have validated the Ekman spiral through profiles showing wind veering and speed reduction near the surface, often under steady synoptic conditions. Post-2000 advancements in Doppler LIDAR technology have enabled high-resolution measurements in urban boundary layers, capturing coherent secondary circulations such as streamwise rolls and their spatial variability over heterogeneous terrain. For example, LIDAR data from convective periods reveal these structures' role in modulating wind gusts and momentum fluxes, with vertical velocities up to several meters per second in organized features.^[27]^[28]

Tropical Cyclones and Vortices

In tropical cyclones, the secondary circulation manifests as a radially inward and upward flow near the surface, transitioning to outward flow aloft, which sustains the primary tangential winds through angular momentum transport. This circulation features strong updrafts in the eyewall, where air ascends rapidly, and subsidence in the surrounding regions, contributing to the storm's warm core structure.^[29] The Rossby number for these systems is typically on the order of 10 or greater, indicating that inertial forces often dominate in the inner core while rotation remains important on larger scales.^[30] The secondary circulation is driven by diabatic heating from condensation in the eyewall, inducing radial inflow at low levels to compensate for mass divergence and fueling the updraft. In intense cyclones, vertical velocities in the eyewall can exceed 10 m/s, with radial inflows reaching 20-30 m/s near the surface, enhancing storm intensity. Recent satellite observations from the 2020s, including infrared and microwave imagery, have enabled estimation of secondary circulation strength via upper-level divergence patterns, correlating these metrics with rapid intensification events—for instance, linking enhanced outflow jets to sustained wind speeds over 50 m/s in systems like Typhoon Haishen (2020).^[31]^[32]^[33] Tornadoes represent smaller-scale, highly concentrated vortices where secondary flows dominate the dynamics, featuring intense radial convergence and vertical updrafts that tilt and stretch ambient vorticity into the vertical. These secondary velocities can reach up to 50 m/s, primarily driven by buoyancy from convective updrafts and the tilting of horizontal vorticity generated by wind shear. In tornado-like simulations, the secondary circulation often features a two-celled structure with intense radial inflow and vertical updrafts that stretch vorticity, supporting the vortex's persistence.^[34]^[35] Dust devils serve as smaller atmospheric analogs to these vortices, initiated by thermal forcing from surface heating that creates buoyant updrafts and secondary circulations on scales of 10-100 m in diameter. The secondary flow involves radial inflow of near-surface air into a low-pressure core, spiraling upward with vertical speeds of 5-20 m/s, lofting dust and debris while dissipating heat through mixing. These transient features highlight the role of localized convection in generating rotation-dominated flows without large-scale organization.^[36]^[37]

Hydrological Examples

Meandering Rivers

In meandering rivers, secondary flows arise primarily from channel curvature, inducing a helical flow pattern that redistributes the primary streamwise velocity across the channel cross-section. This pattern features near-surface flow directed toward the outer (concave) bank and near-bed flow toward the inner (convex) bank, driven by the imbalance between centrifugal forces and cross-stream pressure gradients. The secondary velocity magnitude is typically on the order of 5-20% of the primary flow velocity, significantly influencing flow structure even at low relative intensities.^[38]^[39]^[40] The helical secondary flow promotes geomorphic changes by enhancing outer bank erosion, where high-velocity near-bed flow directed outward increases shear stress and sediment entrainment. Conversely, on the inner bank, the inward near-bed secondary flow reduces velocities, fostering sediment deposition and point bar formation. This process is amplified by cross-stream shear generated by the velocity gradient, which drives transverse sediment transport and links erosion at the outer bank to accretion on inner point bars. Bankfull discharge, representing the channel-forming flow, intensifies these effects by increasing overall flow momentum and secondary circulation strength, particularly in bends with high curvature. Secondary flows become notably stronger in sharp bends with radius of curvature to channel width ratios below 3-5, leading to sharper velocity redistribution and heightened erosional potential.^[39]^[41]^[42]^[43] Flume experiments from the 2010s have quantified how secondary cells modulate turbulence in meandering bends, revealing interactions that alter bed shear stress and sediment dynamics. For instance, large eddy simulations (LES) in 90° and 180° bends demonstrated that outer-bank secondary cells persist longer than predicted by simpler models, enhancing turbulence anisotropy and promoting localized scour. These studies also showed that turbulence driven by secondary flows contributes to significant variations in bed shear stress, directly influencing sediment transport rates and point bar evolution in sharp bends. Recent numerical modeling as of 2024 further confirms these patterns through combined experimental and computational approaches in varied channel configurations.^[44]^[45]^[46] Such findings underscore the role of turbulence modulation in sustaining meander migration under bankfull conditions.

Oceanic Eddies

Oceanic eddies represent a prominent manifestation of secondary flows in the marine environment, where large-scale rotations and density gradients induce ageostrophic circulations that deviate from the dominant geostrophic balance. In frontal zones, such as those associated with ocean currents, geostrophic adjustment processes generate secondary circulations to restore balance after perturbations, featuring ageostrophic components typically comprising 1-5% of the primary geostrophic velocities. These components manifest as cross-frontal flows, with horizontal magnitudes around 2 cm/s compared to along-frontal geostrophic speeds of ~0.5 m/s, driven by the need to counteract imbalances in thermal wind relations.^[47] Vertical velocities in these adjustments can reach ~20 m/day, facilitating the restratification of the water column and influencing tracer distributions across the front.^[47] Mesoscale eddies, scaling with the Rossby deformation radius (typically 10-100 km in the ocean), exhibit secondary flows arising from curvature effects and strain fields that enhance mixing and material transport. In structures like Gulf Stream warm-core rings, these secondary velocities drive radial inflows and outflows, with magnitudes on the order of 0.1 m/s contributing to the dissipation of eddy energy and nutrient upwelling.^[48] The secondary circulation often forms closed cells within the eddy, where centrifugal forces in the azimuthal primary flow induce inward or outward radial components, promoting vertical exchanges that sustain the eddy's longevity and impact regional biogeochemistry. Such flows are crucial for the global oceanic mixing, as they bridge the gap between geostrophic mesoscale dynamics and smaller-scale turbulence. Density-driven effects further amplify secondary flows in oceanic eddies through baroclinic instabilities, where horizontal density gradients along fronts destabilize the flow, generating cross-frontal circulations. These instabilities release available potential energy into kinetic energy, producing meandering fronts and detached eddies with associated ageostrophic secondary cells that converge or diverge across the front, enhancing subduction or upwelling.^[49] In baroclinic fronts, this process leads to turbulent mixing in the surface boundary layer, with cross-frontal eddy fluxes correlating with wave activity and driving significant water mass transformations.^[49] Observational advances since the 2000s, particularly from the Argo float array, have illuminated the global role of secondary motions in eddy kinetics by providing full-depth profiles that enable estimation of eddy kinetic energy (EKE) distributions. Argo data reveal that mesoscale EKE peaks in western boundary current regions, with secondary ageostrophic components contributing to the vertical structure and energy transfers within eddies, accounting for enhanced dissipation and mixing globally.^[50] These observations, spanning over two decades as of 2023, quantify how secondary flows modulate the total EKE, which reaches up to 10^3 cm²/s² in intense eddy fields, underscoring their importance in the ocean's energy budget and climate regulation. Ongoing Argo enhancements through 2025 continue to refine these estimates with improved spatial coverage.^[50]

Engineering Applications

Flow in Containers and Stirring

Secondary flows in simple containers and stirring setups serve as fundamental laboratory models for understanding Ekman-driven circulations in rotating fluids. These experiments typically involve cylindrical vessels filled with viscous fluid, subjected to rotation either by stirring or by rotating the container itself, revealing secondary meridional circulations that arise from boundary layer dynamics at the endwalls. A classic illustration is the tea leaf paradox, where heavy particles like tea leaves collect at the center of the bottom of a stirred cup despite the expectation of centrifugal expulsion to the periphery. Albert Einstein first explained this in 1926, attributing the phenomenon to frictional drag at the cup's bottom, which induces a secondary circulation that transports sediment inward.^[51] The key mechanism is the formation of an Ekman layer—a thin viscous boundary layer of thickness \delta \approx \sqrt{2\nu / \Omega}, where \nu is the kinematic viscosity and \Omega is the angular rotation rate—at the bottom surface. Within this layer, the azimuthal velocity lags behind the bulk rotation due to no-slip conditions, creating a radial pressure gradient that drives an inward radial flow near the bottom. This is counterbalanced by an outward radial flow in the bulk fluid, establishing a single toroidal secondary circulation cell that slowly advects settling particles to the center.^[52] The strength of this secondary flow is characterized by a radial velocity scaling u_r \sim \sqrt{\nu \Omega}, independent of the container radius for large aspect ratios, as derived from Ekman layer balances in rotating fluid theory.^[53] This scaling highlights the role of viscosity in setting the flow intensity, with typical velocities in water ( \nu \approx 10^{-6} m²/s) at moderate stirring rates ( \Omega \approx 1 rad/s) on the order of millimeters per second, sufficient to dominate particle transport over primary rotational motion. In broader cylindrical geometries, such as closed rotating bowls or cups, endwall Ekman layers interact with sidewall boundary layers, potentially leading to multi-cell circulation patterns beyond the simple single-cell regime. For instance, differential rotation between endwalls and the sidewall can generate Stewartson layers along the cylindrical wall, organizing the flow into multiple meridional cells driven by centrifugal instabilities and viscous pumping.^[54] These patterns emerge at higher rotation rates or aspect ratios, where the secondary flow transitions from axisymmetric to more complex structures, enhancing mixing efficiency in stirred systems. Laboratory demonstrations frequently employ dye visualization to capture these secondary flows, with dye streaks injected near the boundaries and azimuthally averaged to reveal the meridional circulation planes. This technique clearly delineates the inward bottom flow and return upper flow in the tea leaf setup, or multi-cell divisions in taller cylinders, providing intuitive evidence of Ekman pumping.^[52] Particle image velocimetry (PIV) studies from the 1990s further validated these models by quantifying secondary velocities in rotating containers, confirming the \sqrt{\nu \Omega} scaling and visualizing azimuthal averaging of meridional components with sub-millimeter resolution. For example, early PIV applications in differentially rotating cylinders measured radial inflows matching theoretical predictions within 10-20%, bridging the gap between analytical Ekman solutions and experimental observations in controlled setups.^[54]

Turbomachinery and Propulsion Systems

In axial compressors, tip leakage vortices form as secondary flows due to the pressure difference across the blade tips, where fluid leaks from the pressure side to the suction side through the clearance gap, rolling up into a coherent vortex structure that migrates toward the blade suction surface.^[55] This vortex interacts with the main passage flow, generating additional mixing losses and reducing the overall isentropic efficiency by 5-10% in typical designs, primarily through increased total pressure loss and blockage effects that exacerbate stall margins.^[56] To mitigate these losses, casing treatments—such as circumferential grooves or slots above the rotor tips—are employed to recirculate high-momentum fluid into the tip region, weakening the vortex strength and recovering up to 2-3% in efficiency while extending the stable operating range.^[57] Experimental and computational studies confirm that optimal groove geometries, often with axial slots, reduce tip leakage mass flow by 20-30% without significantly increasing parasitic drag.^[58] In gas turbine combustors, swirl-induced secondary flows play a critical role in flame stabilization by generating central and corner recirculation zones that enhance mixing of fuel and air while recirculating hot combustion products to anchor the flame front.^[59] The swirl, typically imposed via radial or axial vanes, creates a low-pressure core that draws in reverse flow, forming a toroidal vortex structure with velocities up to 20-30% of the inlet speed, which sustains ignition under lean premixed conditions and reduces NOx emissions through moderated temperature profiles.^[60] These recirculation zones, however, can introduce unsteady oscillations if swirl numbers exceed 0.6-0.8, leading to thermoacoustic instabilities, though advanced designs with variable swirl profiles maintain stability across part-load operations.^[61] In air-breathing propulsion systems, secondary injection flows are strategically introduced perpendicular or at angles to the primary supersonic stream to enhance fuel mixing and thrust vector control, particularly in ramjets and scramjets where residence times are limited to milliseconds.^[62] For supersonic ramjets, oblique shocks generated by wedges or ramps induce cross-flow components that interact with injected fluids, promoting vortex formation and improving penetration depths by 15-25% compared to non-shocked cases, though this also amplifies boundary layer separation and total pressure losses.^[63] In scramjet configurations, these shock-induced secondary flows pose significant challenges in hypersonic environments (Mach 8+), as rapid mixing is hindered by strong shear layers and unstart risks from excessive cross-flow deflection, with recent research emphasizing cavity flameholders to contain recirculation and achieve combustion efficiencies above 90% under 2020s test conditions.^[64] Performance impacts of secondary flows in turbomachinery are quantified using Dean number correlations, which characterize curvature-driven instabilities in blade passages analogous to coiled ducts, where the parameter De = Re \sqrt{\frac{r}{R}} (with Re as Reynolds number, r hydraulic radius, and R curvature radius) predicts the onset of paired counter-rotating vortices that augment heat transfer but elevate losses by 10-20% at De > 100.^[65] These correlations guide blade passage designs to minimize endwall migration of low-momentum fluid, reducing secondary kinetic energy dissipation. Post-2010 computational fluid dynamics (CFD) optimizations, leveraging high-fidelity large eddy simulations and machine learning surrogates, have enabled iterative redesigns that suppress vortex breakdown, yielding efficiency gains of 1-4% in multistage compressors and turbines through adjoint-based shape perturbations.^[66] Such advancements address the compounded losses from secondary flows, which contribute up to 30% of total aerodynamic inefficiencies in modern engines.^[67]

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While viscous flow is laminar in straight pipes, it is not so in curved pipes, because oC the secondary currents which result from the curvature. Page 3 ...
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Prandtl's Mixing-Length Theory - an overview | ScienceDirect Topics
Prandtl mixing length theory refers to a model that treats kinematic viscosity in a manner similar to molecular viscosity, allowing for the determination of the ...
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Experiments on stream-line motion in curved pipes - Journals
Experiments showed that even small curvatures in pipes affect water flow, especially below critical velocity, and slight curves lessen flow.<|control11|><|separator|>
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[PDF] chapter 3 governing equations of fluid dynamics - DICCA
3.3 Transformation of the Governing Equations to Generalized. Curvilinear Coordinates. The Navier-Stokes system of equation (eq. 3.1, eq. 3.2, eq. 3.3 and eq ...Missing: secondary | Show results with:secondary
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[PDF] Fronts, Ekman Boundary Layers and Vortex Flows
» Observations show that frictional effects in the atmospheric boundary layer extend through a depth of about 1 km. » Assume (crude!) that turbulent ...
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[PDF] Planetary Boundary Layer Turbulence - ucla.edu
the turbulent Ekman layer has a less pronounced Ekman spiral than the laminar solutions with the turning angle for the near-surface wind, ≈ 29o, as ...<|separator|>
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Ekman Layer - an overview | ScienceDirect Topics
The observed thickness of the atmospheric boundary layer is of order 1 km, which implies an eddy viscosity of order νV ∼ 50 m2/s.
[24]
Longitudinal instabilities and secondary flows in the planetary ...
The flow in the atmospheric planetary boundary layer is often organized into helical secondary circulations aligned parallel to the mean flow.
[25]
Measurements of Boundary Layer Profiles in an Urban Environment in
The lidar-derived profiles are robust and accurate even for challenging conditions such as stable boundary layers with a low-level jet, low turbulence, and low ...
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Convectively Induced Secondary Circulations and Wind‐Driven ...
Oct 22, 2024 · Lidar and surface wind measurements provide evidence linking widely observed secondary circulations to surface wind gusts The circulations ...
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Hurricane Dynamics - NOAA
of a hurricane in the radius-height plane shows that the primary swirling flow is maintained by a radial and vertical secondary circulation. Under the ...
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[PDF] Tropical Cyclone Structure - Meteorology
• The first term is the Rossby number, RO. ○ In the core of the vortex, the Rossby number is large, so the Coriolis effects can be ignored. Therefore, the ...
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Diabatically Induced Secondary Flows in Tropical Cyclones. Part I
Secondary circulation induced by the specified heat source acting on the idealized vortex: (a) mass flow streamfunction (kg s21), (b) vertical velocity (m s21), ...
[30]
Divergence - CIMSS Tropical Cyclones
Uses: Upper-level atmospheric divergence can be used by tropical cyclone (TC) forecasters to estimate the strength of the TC secondary circulation, including ...
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Case Study for Typhoon Haishen (2020) with Atmospheric Motion ...
The method was applied to the 30-s special observation of Typhoon Haishen (2020) with the Himawari-8 satellite. Various screening was applied to secure quality.
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[PDF] The Fluid Dynamics of Tornadoes - NCAR/MMM
Sep 17, 2012 · Wind-speed limits in numerically simulated tornadoes ... Three-dimensional instabilities in tornado-like vortices with secondary circulations.
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The Three-Dimensional Structure and Evolution of a Tornado ...
Colored contours depict the tangential velocity Vt, vectors depict the secondary circulation (Vr, w), and white line contours depict the angular momentum ...
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Dust devil dynamics - Horton - 2016 - AGU Journals - Wiley
May 12, 2016 · A self-consistent hydrodynamic model for the solar heating-driven onset of a dust devil vortex is derived and analyzed.
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A Simple Thermodynamical Theory for Dust Devils in - AMS Journals
The theory uses a heat engine framework to scale dust devils, predicting their intensity, diurnal variation, and spatial distribution, based on thermodynamic ...
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A Review of Numerical Simulations of Secondary Flows in River ...
This study comprehensively reviews the fundamentals of secondary flow, the governing equations and boundary conditions for numerical simulations,
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Flow of water in bends of open channels - Semantic Scholar
Laboratory Study of Secondary Flow in an Open Channel Bend by Using PIV · Engineering, Environmental Science. Water · 2019.
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https://www.isws.illinois.edu/pubdoc/CR/ISWSCR-400.pdf
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The influence of geostrophic strain on oceanic ageostrophic motion ...
Jun 28, 2019 · In order to separate the signals of ageostrophic motions, the ocean surface ageostrophic velocity is computed as ua = u – ug – u0, where ug is ...
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Warm Spiral Streamers over Gulf Stream Warm-Core Rings in
Abstract. This study examines the generation of warm spiral structures (referred to as spiral streamers here) over Gulf Stream warm-core rings.
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Baroclinic Frontal Instabilities and Turbulent Mixing in the Surface ...
For fronts with significant baroclinic wave activity, cross-frontal eddy fluxes computed from correlations of fluctuations from means along the large-scale ...
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Full‐Depth Eddy Kinetic Energy in the Global Ocean Estimated From ...
Aug 4, 2023 · Key Points · A new method is developed for estimating full-depth eddy kinetic energy (EKE) from satellite altimeter and Argo float data.Missing: secondary ageostrophic
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Die Ursache der Mäanderbildung der Flußläufe und des ...
Einstein, A. Die Ursache der Mäanderbildung der Flußläufe und des sogenannten Baerschen Gesetzes. Naturwissenschaften 14, 223–224 (1926).
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https://www.nature.com/articles/s41467-019-10883-w
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[PDF] Nonlinear Dynamics II: Continuum Systems, The Ekman Layer
Thus, the flow in the inner region has a velocity which is entirely set by the boundary layers. Note that there is no viscosity in this formula, but viscosity ...
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Characteristics of endwall and sidewall boundary layers in a rotating ...
Mar 25, 1998 · The secondary induced meridional flow is also potentially unstable in this region. This is manifested by the inflectional radial profiles of the ...
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Axial compressor blades tip leakage flow control using ... - Nature
Dec 19, 2023 · Generally speaking, the tip leakage flow losses may constitute about 20–30% of the total losses. Of pioneers who have studied on the tip leakage ...Missing: percentage | Show results with:percentage
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Compressor Tip Leakage Mechanisms | J. Turbomach.
1 Introduction. Tip leakage losses in axial compressors are significant and account for up to a third of the total loss [1]. Losses occur as the blade pressure ...Introduction · Multi-Order Methodology · Loss Mechanisms · Reduced Loss BladingMissing: efficiency percentage
[52]
Casing Treatment: Its Potential and Limitations | GT
Oct 28, 2022 · Casing treatment is an advanced design feature intended to improve the stability of a compressor. Various investigations have been conducted.
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Multi-Objective Optimization of Circumferential Groove Casing ...
Mar 22, 2022 · The tip flow filed analysis shows that the optimized grooved casing treatment can suppress the shock/TLV interaction and double-leakage flow.
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Flow and Flame Mechanisms for Swirl-Stabilized Combustors - MDPI
Swirl is utilized in reacting flows to ensure the robust stabilization of the flame because swirl induces an inner recirculation zone of high temperature and a ...
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Effects of unstable flame structure and recirculation zones in a swirl ...
This study addresses structural characteristics of natural gas flames in a lean premixed swirl-stabilized combustor with attention focused on the effect of the ...
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[PDF] Effect of Air Swirler Configuration on Lean Direct Injector Flow ...
Some unsteadiness may be caused by swirling flow, which is typically used in gas turbine combustors in order to establish a central recirculation zones (CRZ).Missing: induced secondary
[57]
Research Progress on Active Secondary Jet Technology in ... - MDPI
The active secondary jet is capable of changing the flow field structure by injecting one or more secondary streams into the primary flow at a specific velocity ...
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Influence of oblique shock interaction on the liquid jet in supersonic ...
This study shows that oblique shock interacting with the liquid jet in supersonic crossflow enhances the penetration.
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Research progress of the flow and combustion organization for the ...
May 1, 2025 · This paper delves into the intricacies of the reacting flows within high-Mach-number scramjets through a comprehensive literature review.
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Fluid Flow and Heat Transfer in Rotating Curved Duct at High ...
The internal cooling channels of gas turbine blades usually consist of two or more straight passages connected with 180 deg bends. As the fluid reaches the bend ...
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Machine Learning Methods in CFD for Turbomachinery: A Review
This paper illustrates recent developments in CFD for turbomachinery which make use of machine learning techniques to augment prediction accuracy.
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Numerical Investigation of Secondary Flow and Loss Development ...
In current axial turbines, secondary losses account for approximately 30 % of the overall losses (Cui and Tucker [2]). Whereas early studies on secondary flows ...