Fig. 1 illustrates the optimized three-dimensional geometric model of the negative-pressure yarn-end detection and adsorption system, which integrates four functionally coordinated components: a yarn package spindle assembly for stable mounting, a vacuum cantilever for securing the suction nozzle, a three-axis positioner for precise nozzle adjustment, and a negative-pressure suction nozzle serving as the terminal actuator. The system operates through a precisely controlled sequence, wherein the vacuum pump generates a regulated negative-pressure jet flow that is directed through the suction nozzle to interact with the yarn package surface. At the same time, a rotary drive mechanism drives the nozzle to rotate around the yarn package. By synchronizing the rotation of the nozzle with the pressure, the system finds the yarn end stably and reliably.

Based on the experimental setup, a geometric model of the suction nozzle was established. The experimental yarn-end motion was then reproduced by the model with higher accuracy. The requirements of the previous process design and the common configuration of textile suction nozzles reported in the literature were integrated to determine the key structural parameters of the nozzle. The total length of the nozzle was l = 140 mm. As illustrated in Fig. 1, the nozzle consisted of three parts: the throat, the divergent conical section, and the outlet segment. The length of the throat was l_n = 12 mm, and its outer diameter was d = 12 mm, where the negative-pressure pipeline was connected. The length of the conical section was about l_s = 90 mm, and the divergence angle was α = 25°. The conical section was shaped to suppress the flow separation and to form a smooth streamline transition. The outlet section has a nozzle length of L = 90 mm, for a yarn package diameter D_p = 100 mm, the nozzle width was set as d = D_p/5 = 100 mm ensuring that the yarn end can smoothly enter the airflow channel without forming recirculation zones or vortex attachment.

To define the boundary conditions of the negative-pressure jet model more accurately, a computational fluid domain with dimensions of 100 mm, 30 mm, and 40 mm in the X, Y, and Z directions, respectively, was constructed based on the simplified suction-nozzle geometry. The air inlet at the end of the nozzle was specified as the inlet boundary, while the bottom opening of the nozzle together with the outer surfaces of the computational domain were set as outlet boundaries. All remaining surfaces of the nozzle were treated as wall boundaries. An unstructured tetrahedral mesh was employed to discretize the flow field.

The airflow within the negative-pressure suction nozzle and over the yarn package must satisfy the fundamental governing equations of fluid mechanics, namely the continuity, momentum, energy, and state equations [31]. The nozzle inlet diameter at the top is 12 mm, where the negative-pressure jet velocity can reach up to 240 m/s. Under the present operating conditions, the airflow inside the nozzle is regarded as a three-dimensional, viscous, compressible (density-varying) turbulent flow. Considering both accuracy and computational cost, the present study adopts the RANS approach [32].

According to the law of conservation of mass, the expression of the vector form of the fluid continuous equation is:

In this equation, ρ , t, and u denote the air density, time, and the airflow velocity vector, respectively, and ∇ is the nabla operator.

The momentum equation per unit mass for a fluid can be derived from conservation laws and expressed in vector form as: (2) D u D t = f − 1 ρ ∇  p − 1 ρ ∇ 2 3 μ e f f ∇ · u + 1 ρ ∇ · 2 μ e f f S − 1 ρ ∇ 2 3 ρ k where D u D t is the acceleration of an air fluid particle (material acceleration). The term f denotes the body-force acceleration acting on the fluid particle, in the present study gravity is neglected and thus f = 0 . The term − 1 ρ ∇ p represents the pressure-gradient force per unit mass. The term − 1 ρ   ∇ 2 3   μ e f f   ∇ · u is the dilatational (bulk-viscous) contribution associated with volumetric deformation, while 1 ρ ∇ · 2 μ e f f S is the divergence of the deviatoric viscous stress tensor (shear-viscous contribution). The term − 1 ρ   ∇ 2 3 ρ   k accounts for the isotropic part of the turbulent Reynolds stress introduced. Here, S = 1 2 ∇   u + ∇   u T is the mean rate-of-strain tensor and k is the turbulent kinetic energy.

To incorporate the turbulence-enhanced momentum diffusion in the RANS framework, an effective viscosity is introduced as (3) μ e f f = μ + μ t where μ is the molecular dynamic viscosity of air, and μ t is the turbulent (eddy) viscosity, which represents the contribution of turbulent fluctuations to the mean stress. The eddy viscosity μ t is evaluated by the RNG k-ε turbulence model.

The energy equation is derived by applying the first law of thermodynamics to a fluid control volume:

In the formula, the term k e f f represents the effective thermal conductivity of the fluid, and T is the temperature. The term q ˙ denotes the volumetric heat generation rate, while ρ f · u represents the work (power per unit volume) done by external body forces on the fluid. In the present study, no volumetric heat source is considered, so q ˙ = 0 , and gravity is neglected, so f = 0 and thus ρ f · u = 0 .

As illustrated in Fig. 2, the nozzle has a narrow inlet followed by a conical diverging section, where the airflow accelerates sharply near the entrance and then decelerates downstream. To capture this high-strain-rate turbulent flow, the RNG k-ε model [33] is employed. In the RANS framework, turbulence closure is obtained by solving the transport equations for k and ε from which the eddy viscosity μ t is computed and used in the momentum equation. The governing equations of the RNG k-ε model are given as follows: (5) ∂ ρ k ∂ t + ∇ · ρ k u = ∇ · μ + μ t σ k ∇ k + G k − ρ ε (6) ∂ ρ ε ∂ t + ∇ · ρ ε u = ∇ · μ + μ t σ ε ∇ ε + C 1 ε ε k G k − C 2 ε ρ ε 2 k − R ε where G k is the production of turbulent kinetic energy due to the mean velocity gradients. The eddy viscosity is computed as

In the RNG formulation, an additional correction term R ε is introduced in the ε -equation to improve the sensitivity to rapidly strained and swirling flows, and it is expressed as (8) R ε = C μ ρ η 3 1 − η / η 0 ε 2 k 1 + β η 3 , η = S k ε where R ε is the additional RNG correction term defined in Eq. (8), in which η = S k ε and S is the magnitude of the mean rate-of-strain tensor. The model constants were taken as σ k = 0.7194 , σ ε = 0.7194 , C μ = 0.0845 , C 1 ε = 1.42 , C 2 ε = 1.68 , η 0 = 4.38 , and β = 0.012, following the standard RNG k-ε formulation as implemented in ANSYS Fluent.

Since the above governing equations involve thermodynamic variables, an equation of state is required to close the system. In this study, air is treated as an ideal gas, and the density is related to pressure and temperature by (9) p = RρT = γ − 1 ρe where R is the molar gas constant, c p is the specific heat capacity at constant pressure, c v is the specific heat capacity at constant volume, and γ = c p / c v is the specific heat ratio.

To evaluate the influence of grid resolution on the numerical predictions, a mesh sensitivity analysis was further conducted. Three meshes were generated for the same computational domain, namely Mesh A (coarse), Mesh B (medium), and Mesh C (fine), comprising approximately 1.00 × 10⁶, 2.76 × 10⁶, and 4.07 × 10⁶ cells, respectively. Under identical boundary conditions and turbulence-model settings, simulations were performed for two representative operating conditions. The plane-averaged velocity and plane-averaged total pressure were then evaluated on three characteristic cross-sections located at Y = 20, 60, and 120 mm. The quantitative results are summarized in Table 1.

The differences among the three meshes are limited: the maximum relative deviation is 4.17% for the mean velocity and 4.79% for the mean total pressure, both below 5%. Accordingly, the airflow predictions can be regarded as nearly mesh-independent for the present purpose, and the medium mesh was adopted for subsequent simulations to balance accuracy and computational cost.

Given that cotton yarn is the most widely used material in the textile industry due to its excellent spinnability, flexibility, and aerodynamic properties. A 55 tex pure cotton yarn was selected as the research subject in this study. A 50 m sample was prepared, and its mass was measured using an analytical balance. The equivalent diameter of the yarn cross section was determined through scanning electron microscopy (SEM), from which the yarn density was calculated. The stress–strain curve of the yarn was obtained using an Instron 3345 universal testing machine, and the Young’s modulus was derived by linear fitting. The corresponding physical parameters are summarized in Table 2. Based on the data listed in Table 2, a solid model of the yarn was subsequently constructed.

In the present simulations, the yarn length was set to L = 10 mm. Based on the yarn density and diameter, the corresponding weight is estimated to be G ≈ 0.0054 mN. By contrast, the subsequent results show that the aerodynamic force acting on the yarn during the capturing stage is typically on the order of 3–9 mN (with a maximum of about 9.149 mN), yielding Fa/G~10²–10³. Moreover, the main action direction of the negative-pressure jet is approximately perpendicular to gravity, which further weakens the influence of gravity on the suction-driven axial motion and the early-stage capturing dynamics. These order-of-magnitude considerations support neglecting gravity for the present capturing process in which aerodynamic effects are dominant. In addition, yarn twist and surface roughness (hairiness) mainly affect secondary details through their influence on local drag characteristics and yarn–wall contact friction. For the present problem, where the transient capture of the free yarn end is governed primarily by suction intensity and flow-field structure, these effects can be regarded as of second order. Considering the difficulty of highly realistic modeling and the associated computational cost, gravity as well as yarn twist and surface roughness were neglected in the CFD simulations to reduce the overall complexity.

It should be noted that the above simplifications appear appropriate for the high-speed, aerodynamically dominated capture stage examined in this work. However, the findings could indicate that under low-velocity conditions or during prolonged near-wall motion, gravity might affect the sagging configuration and contact posture of the yarn. Moreover, neglecting twist means that its influence on the effective bending stiffness and lateral aerodynamic response is not represented. Ignoring surface roughness (hairiness) may lead to an underestimation of frictional resistance and adhesion tendency at the yarn–wall interface. Nevertheless, the present model is intended to describe the overall transient capture behavior of a free yarn end under negative-pressure suction rather than to resolve detailed microscopic contact processes. Thus, within the scope and objectives of this study, these assumptions are considered acceptable.

Because the yarn is thin and long, it possesses high flexibility while retaining a certain degree of stiffness. This implies that three-dimensional modeling of the yarn must simultaneously account for its displacement and deformation, as well as the axial forces, bending moments, and torques exerted on it within the jet flow field [34]. It has been found that yarn models constructed using conventional truss or bar elements are highly prone to numerical instabilities during simulation, which often result in non-physical distortions, entanglements, or self-intersections of the yarn structure, making it difficult to accurately capture the real mechanical behavior of yarns. To overcome this issue, the present study employs a three-dimensional yarn model based on beam elements, which effectively eliminates the aforementioned twisting problem and exhibits good numerical stability. Ultimately, the yarn solid model was developed using a digital chain discretization approach, which involves continuous elastic spatial beam elements, as illustrated in Fig. 2.

In the present Abaqus implementation, the yarn is represented by an equivalent linear-elastic constitutive model for the beam elements. Moreover, the material parameters may be identified from the linear portion of the measured tensile stress-strain response, giving E = 80 MPa, ν = 0.307, and ρ = 485 kg/m³ (Table 2). A time-dependent viscoelastic description (e.g., Prony-series relaxation/creep or rate-dependent effects) is not included. Given that yarn-end capture occurs on a millisecond time scale and is primarily governed by aerodynamic loading, viscoelasticity is expected to have a limited first-order influence on the overall transient trajectory and adsorption time; therefore, the equivalent elastic approximation is adopted to enhance reproducibility while keeping the computational cost tractable.

In addition, a discretization sensitivity check was performed for the beam-element yarn model by varying the number of beam elements. The predicted yarn trajectory and capture-related metrics showed no noticeable change, indicating that the adopted discretization (500 beam elements) is sufficiently fine for the present simulations.

In this study, the suction level was varied from −0.04 to −0.07 MPa. The yarn used in the experiments was a 55 tex cotton yarn, which was represented by beam elements (Table 2). Within this range, the yarn diameter is much smaller than the characteristic size of the negative-pressure suction nozzle, and the volume fraction in the flow domain is negligible. Therefore, the feedback of yarn deformation on the main flow structure inside the nozzle is expected to be limited. Moreover, the dominant features of the internal jet are governed by the nozzle geometry and the prescribed negative-pressure boundary condition. Based on these scale considerations and the dominant physical mechanisms, a weakly coupled FSI strategy [35] was adopted to solve the airflow loading and the yarn response, achieving efficient and flexible computation while maintaining adequate accuracy. In the present work, this weakly coupled strategy is implemented as a one-way coupling approach: a steady RANS airflow field is solved first and then used to drive the yarn motion, while the influence of yarn deformation on the airflow field is neglected. However, when the suction level is increased substantially, when a more compliant yarn may lead to pronounced lateral oscillations, or when persistent wall contact causes noticeable modification of the local flow, the fluid–structure feedback may become stronger. In such cases, a fully coupled (two-way) FSI treatment may be required to further improve predictive accuracy.

The negative-pressure jet inside the suction nozzle was computed in ANSYS Fluent using the pressure-based steady solver. Because a steady flow formulation is adopted, the SIMPLE algorithm (rather than PISO) was used for pressure–velocity coupling. Spatial discretization employed least-squares cell-based gradient evaluation; the pressure term was discretized using the second-order scheme, and the momentum as well as the RNG k and ε transport equations were discretized using the second-order upwind scheme (second-order accuracy). Convergence was assessed by monitoring scaled residuals and integral quantities (e.g., mass flow rate and pressure level) until stable values were reached. These numerical settings are consistent with Fluent-based CFD practice reported in FDMP studies employing the pressure-based steady solver with SIMPLE coupling and second-order discretization schemes [36,37].

The negative-pressure jet inside the suction nozzle is solved using the Fluent fluid solver, while the yarn end is modeled as a digital chain under the elastic beam framework, which requires the Abaqus solver to compute the spatial motion and configuration of the yarn. To enable data exchange between the two solvers, a Java program is developed to invoke the inverse distance weighting (IDW) interpolation algorithm embedded in Tecplot, thereby extracting the fluid data at the beam-element nodes of the yarn. Subsequently, the extracted fluid data are transformed into the corresponding aerodynamic forces at the yarn nodes using the aerodynamic force formula derived from the Buckingham π theorem. Finally, the yarn’s displacement and deformation are acquired in Abaqus by employing an explicit dynamic algorithm and updating the results during the calculation.

For the numerical implementation, the three-dimensional yarn model was discretized into a digital chain composed of elastic beam elements. However, this discretization also leads to a spatial mismatch between the yarn and the negative-pressure suction flow field’s fluid mesh, because the nodal positions of the yarn and fluid mesh seldom coincide. Nevertheless, a certain degree of spatial correlation still exists between the beam-element nodes of the yarn and the surrounding fluid elements. To address this, the inverse distance weighting (IDW) interpolation algorithm is employed, taking the beam-element node as the interpolation center. Within a prescribed search range, the fluid parameters at each yarn node are estimated from the known fluid element data based on the principle of distance-weighted averaging, the interpolation procedure is illustrated in Fig. 3.

By applying the following formula, a quantitative connection between the surrounding airflow field and the yarn nodes is established, thereby obtaining the airflow data at any given node in the yarn entity model.

In the formula group, where Z p is the interpolated result of the flow-field variable (e.g., velocity, aerodynamic force, etc.) at the yarn beam element node p, Z i represents the value of the flow-field variable at the i-th fluid mesh node obtained from Fluent simulations, λ i is the weighting coefficient of the i-th fluid mesh sampling point. d i is the Euclidean distance between the i-th mesh sampling point and the interpolation node p. The exponent k controls the rate at which the weight decays with increasing distance.

The yarn’s motion can be described by Newton’s second law; its internal and external forces are depicted in Fig. 4. The internal forces arise from the interaction between neighboring beam elements within the same yarn, which generate axial tension, bending, and torsional moments on the cross-sections of the element. The external force mainly results from the traction of the negative-pressure jet, which is called the aerodynamic force here.

When the negative-pressure jet flows across the yarn surface, the generated aerodynamic force F is primarily governed by three factors: the airflow density ρ , the relative velocity v between the airflow and the yarn, and the effective projected area of the yarn within the airflow A . Based on the Buckingham π theorem, a dimensionless functional relationship among the four quantities F , ρ , v , A can be established, which leads to the derivation of the aerodynamic force. Since the total number of physical quantities is n = 4 and the number of fundamental dimensions is r = 3(M, L, T), a single dimensionless Π -term can be constructed:

By comparing this expression with the general formulation of drag force in fluid mechanics, it can be inferred that the value obtained through the Buckingham π theorem is related to the drag coefficient (C_d). Specifically, the derived value corresponds to one-half of the drag coefficient. Consequently, the expression for the aerodynamic force (F) can be formulated.

Owing to the discretization of the yarn model, the aerodynamic force must be applied to the nodes of each beam element. For this purpose, the total aerodynamic force is decomposed into two components along the yarn’s axial and radial directions: (13) F a = 1 2 C a ρ S a v a v a = 1 2 C a ρπ d ∥ v a ∥ v a (14) F r = 1 2 C r ρ S r v r v r = 1 2 C r ρπ rl ∥ v r ∥ v r where C a and C r denote the axial and radial drag coefficients of the yarn, respectively, S a and S r are the corresponding projected areas of the yarn, v a and v r represent the relative velocities between the yarn and the axial and radial components of the airflow.

Through extensive experimental curve fitting, empirical correlations for the axial and radial drag cotton yarn aerodynamic coefficients have been obtained, which can be expressed as: (15) C a = 0.8945 · | v | − 0.0884 (16) C r = 4.8246 · | v | − 0.5254 here, C a and C r are effective drag coefficients that characterize the resistance of a slender yarn segment to the relative airflow in the axial and radial directions, respectively. They were determined by regressing the measured aerodynamic force against the relative velocity for the same yarn type used in this study. For the axial drag correlation in Eq. (11), the coefficient of determination is R² = 0.977 with a root-mean-square error (RMSE) of 7.8 × 10. For the radial drag correlation in Eq. (12), the fitting yields R² = 0.991 with an RMSE of 4.8 × 10⁻³. These metrics indicate that the proposed correlations reproduce the experimental data well over the velocity levels relevant to the present work, which cover the nozzle-flow conditions where the characteristic velocity reaches up to about 240 m/s. The correlations are therefore directly adopted in the subsequent simulations.

Finally, the nodal displacements of the beam elements of the yarn were acquired by an explicit dynamic simulation. In this scheme, the state variables, such as displacement and velocity, at the end of each time increment are computed from those of the previous step, following the standard explicit integration procedure. In view of the millisecond-scale duration of the yarn-end capture process, a time increment of Δt = 5 × 10⁻⁶ s was adopted for updating the aerodynamic loading in the coupled simulation. No temporal discretization is involved in the CFD solver because a steady airflow solution is computed; the temporal discretization is required only for the yarn dynamics, which are integrated in Abaqus/Explicit using the standard explicit (central-difference) time integration scheme. The time increment was selected to be sufficiently smaller than the millisecond-scale capture time, providing a large number of increments to resolve the transient response while maintaining numerical stability, consistent with explicit time-domain dynamics practice reported in FDMP [38]. With this choice, the transient process can be resolved with a sufficiently large number of time steps, and spurious numerical oscillations in the predicted response are effectively suppressed. Accordingly, the dynamic equilibrium equation of the yarn model is written as: (17) M u ¨ t = F ext t − F int t where u ¨ t denotes the nodal acceleration vector at time t, M represents the lumped mass matrix, F ext t is the external force vector acting on the nodes (including aerodynamic forces and other applied loads), and F int t denotes the internal force vector.

The governing equations of motion are explicitly integrated to evaluate the overall movement of the yarn. (18) u ˙ i + 1 2 = u ˙ i − 1 2 + Δt i + 1 + Δt i 2 u ¨ i (19) u i + 1 = u i + Δt i + 1 u ˙ i + 1 2 where u ˙ and u ¨ denote the yarn velocity and the yarn acceleration, the superscript (i) refers to the simulation increment step, whereas i − 1 2 and i + 1 2 indicate the intermediate values within the increment step.