Simulation of Motion of Long Flexible Fibers with Different Linear Densities in Jet Flow

Air-jet loom is a textile machine designed to drive the long fiber using a combination flow of high-pressure air from a main nozzle and a series of assistant nozzles. To make the suggestion of how to make the fiber fly with high efficiency and stability in the jet flow, in which vortices also have great influence on fiber movement, the large eddy simulation method was employed to obtain the transient flow field of turbulent jet, and a bead-rod chain fibermodel was used to predict long flexible fiber motion.,e fluctuation and velocity of fibers with different linear densities in jet flow were studied numerically. ,e results show that the fluctuation amplitude of a fiber with a linear density of 0.5×10 kg·m is two times larger than that of a fiber with a linear density of 2.0×10 kg·m. ,e distance of the first assistant nozzle from the main nozzle should be less than 120mm to avoid collision between the fiber and the loom. ,e efficient length of the main nozzle to carry the fiber flying steadily forward is about 100–110mm. For fibers with a linear density of 0.5×10 kg·m, it is suggested that the distance of the first assistant nozzle from the main nozzle is about 110mm. With the increase of fiber linear density, the distance could be appropriately increased to 140mm.,e simulation results provide an optimization option for the air-jet loom to improve the energy efficiency by reasonably arranging the first assistant nozzle.


Introduction
Carried flight of a long, flexible fiber in a jet flow field is a fundamental science problem for air-jet loom [1], which is designed to drive the yarn using a combination flow of highpressure air from a main nozzle and a series of assistant nozzles.Different from other pipeline pneumatic conveying systems, the jet flow of the main nozzle will rapidly decelerate in a free area and cannot guarantee a sustained highspeed flight of the fiber.erefore, assistant nozzles must be added to the system.On the other hand, the presence of the reed groove requires that the fluctuation of the fiber must be less than a certain range.How to make the fiber fly forward rapidly and steadily is always a core issue of fiber flight and is jointly determined by two aspects: the fiber model and the flow simulation.
In general, a long fiber can be modeled as a rigid or flexible fiber for different dynamic research purposes.Yamamoto and Matsuoka [2] proposed a method for simulating the dynamic behavior of rigid and flexible fibers in a flow field.e fiber is regarded as made up of spheres that are lined up and bound to each neighbor.Each pair of bonded spheres can stretch, bend, and twist by changing the bond distance, bond angle, and torsion angle between spheres, respectively.For the rigid fiber, the computed period of rotation and distribution of orientation angle agree with those calculated using Jeffery's equation with an equivalent ellipsoidal aspect ratio.For the flexible fiber, the period of rotation decreases rapidly with the growth of bending deformation of the fiber and rotation orbits deviate from a circular one for the rigid fiber.De Meulemeester et al. [3] tried to study the dynamic properties of fiber movement and developed a one-dimensional mathematical model, in which the behavior of the fiber is described by Newton's second law.Lindström and Uesaka [4] proposed a model for flexible fibers in viscous fluid flow; namely, the fibers are modeled as chains of fiber segments which interact with the uid through viscous and dynamic drag forces.Fiber segments, from the same or di erent bers, interact with each other through normal, frictional, and lubrication forces.
e simulations using the proposed model successfully reproduced the di erent regimes of motion for thread-like particles that range from rigid ber motion to complicated orbiting behavior including coiling and self-entanglement.
Vahidkhah and Abdollahi [5] used the lattice Boltzmann method (LBM) to solve the Newtonian ow eld and the immersed boundary method (IBM) to simulate the deformation of the exible ber interacting with the ow.e variations of the ber length during the simulation time for di erent values of stretching constant are studied.Kabanemi and Hétu [6] carried out a direct simulation study to analyze the e ect of ber rigidity on ber motion in simple shear ow.
e ber is modeled as a series of rigid spheres connected by sti springs, which is similar to that used by Yamamoto and Matsuoka [2].e model correctly predicts the orbit period of ber rotation, as well as the trend of critical ow strength, versus ber aspect ratio, during which the breakage occurs in simple shear ow.Meirson and Hrymak [7] extracted rotational friction coe cients from Je ery's model, created a general case long exible ber orientation model, and applied it in a simple shear ow.Nan et al. [8] presented a linear viscoelastic sphere-chain model based on the discrete element method to quantify the material damping of deformed exible bers.A correlation is formulated to quantify the relationship between the damping coe cient of the local bond and that of the exible ber.Meulemeester et al. [9] developed a threedimensional mathematical model of the yarn.
e threedimensional model for the weft insertion on air-jet looms has been successfully tested.
On the other side, as we have mentioned, ber ying in uid is also a ected by turbulent ow characteristics.Kim et al. [10] analyzed the ow in an air-jet loom by using a time-accurate characteristic-based upwind ux-di erence splitting compressible Navier-Stokes method.e unsteady pressure and Mach number behavior along the center line of the main nozzle were analyzed.Andrić et al. [11] analyzed the dynamics of individual exible bers in a turbulent ow, the direct numerical simulation of the incompressible Navier-Stokes equations is used to describe the uid ow in a plane channel, and a one-way coupling is considered between the bers and the uid phase.ey found that the ber motion is primarily governed by velocity correlations of the ow uctuations.In addition, they reported that there is a clear tendency of the thread-like bers to evolve into complex geometrical con gurations in a turbulent ow eld, and the ber inertia has a signi cant impact on reorientation timescales of bers suspended in a turbulent ow eld.Jin et al. [12] simulated the turbulent ow by solving the Reynoldsaveraged Navier-Stokes (RANS) equations and conducted the three-dimensional numerical simulation of the movement of the exible body.
e numerical results show that an unconstrained exible body would turn over forward along the air ow's di usion direction, while a constrained exible body in the ow eld will make a periodic rotation motion along the axis of the exible body, and the bending deformation is more obvious than that of unconstrained exible body.Yang et al. [13] studied the two-way coupling turbulent model and rheological properties for ber suspension in the contraction based on the RANS simulation.
Pei and Yu [14] studied the motional characteristics of the exible bers in the air ow inside the Murata vortex spinning (MVS) nozzle.A two-dimensional uid structure interaction (FSI) model combined with the ber wall contact is introduced to simulate a single ber moving in the air ow inside the MVS nozzle.e model is solved using a nite element code ADINA.Based on their simulation results, the formation principle and the in uence of some nozzle parameters on the tensile property of the MVS ber were discussed.
More and more investigators put their interest on the air-jet loom [15][16][17][18] and other uid machineries [19,20], but due to the computation cost gap between scienti c researches and engineering needs, only a few of them considered the in uence of turbulent uctuation [21].e jet ow caused by the main nozzle is a typical free shear turbulence, and there are strong vortices which play an important role on the momentum and energy transport of ow eld [22,23].e vortices also have great in uence on the ber movement.Large eddy simulation (LES) is a believed mathematical model for turbulence vortex simulation, by directly calculating the large-scale ow motion.
In this paper, we simpli ed the two-phase ow system and were able to employ the LES method to simulate the development of a vortex in jet ow and the Lagrangian beadrod model to give the time evolution of a long exible ber distribution with di erent linear densities.en, the ber uctuation and the velocity were discussed to make the suggestion of how to make the ber y with high e ciency and stability in an air-jet loom.

Fluid Flow.
e 2D jet ow is shown in Figure 1, in which x and y are the streamwise and cross-stream directions, respectively.e width of the nozzle D is 3.5 mm, the uid velocity at the nozzle U 0 is 240 m/s, and the ow Reynolds number Re 5.68 × 10 4 .2 Journal of Nanotechnology e LES equations governing the jet ow obtained by ltering the Navier-Stokes equations are as follows [24]: where ρ is the uid density, u m is the ltered velocity, p is the ltered pressure, index m, n is taken as 1, 2 and refers to the x, y, and μ is the dynamic viscosity.
e SGS (subgrid stress) tensor τ mn and Smargorinsky-Lilly model [24] which are based on the mixing length hypothesis are used to calculate the SGS stress.

Flexible Fiber Model.
A single exible long ber is modeled as a bead-rod chain, which is similar to that used by Guo et al. [25].e ber model is composed of N beads, which are connected by N − 1 massless rods (Figure 2).Only the beads are a ected by forces, and the rods maintain the con guration of the ber.Using the model, the chain is allowed to be stretched by changing the distance of adjacent rods.
If the distance between adjacent beads is not equal to the equilibrium distance, the stretching restoring force F ni exerted on the bead i will be where R is the bead radius, E is the elastic modulus, L is the equilibrium distance, and ΔL is the distance variation, which is equal to the transient distance of each two adjacent beads subtracted by the equilibrium distance.
When immersed in the unsteady ow eld which is calculated in the last section, the ber is subjected to hydrodynamic forces, which are also changed with time.In this paper, only drag force is considered in order to decrease the computational cost.Other hydrodynamic forces such as Basset history term, additional mass, slip-rotational lift force, and uid inertia are negligible.For bead i, the drag force F di is contributed by ber sections (i − 1, i) and (i, i + 1).It can be calculated as follows: where F di−1, i and F di, i+1 are the drag forces acting on bead i, which are devoted to the ber section (i − 1, i) and (i, i + 1), respectively.e drag F di−1, i acting on the ber section (i − 1, i) can be expressed as follows: where V qi and V di are the uid and ber velocities at the mass center of the ber section (i − 1, i), and C d is the drag coe cient associated with Reynolds number, which can be represented as follows: where Re s 2r v ρ|V qi − V 0i | is the Reynolds number of equivalent spherical and β is a constant between 0.4 and 0.45.According to Newton's second law, the equations of motion for the bead i that constitute the ber are as follows: where m i is the mass of the bead i and r i is the position vector of the bead i.

Computation
Conditions. e computational domain covers x × y 21D × 40D, the nozzle width D 3.5 mm, fully developed boundary conditions at the outlet, and static surrounding environment conditions are assumed.e local mesh is shown in Figure 3. Fiber motion is solved in one-way coupling between the ber and ow; ber-wall and berber interactions are neglected.

Jet Flow Field Veri cation.
According to the experimental results of [26,27], the streamwise velocity satis es the self-preservation distribution at x ≥ 8D and the pro le is expressed as a Gaussian curve.e self-preservation pro les of the mean velocity are shown in Figure 4, where the data are normalized by the centerline mean velocity U m and the half-width r 0.5 , which is the distance between the position where the streamwise velocity u/U m 0.5 and the center line of the jet.e gure shows that the air ow which is mentioned in Section 2.1 has been successfully simulated.

Flexible Fiber Model Veri cation.
As the limitation of the visible area and the resolution of the high-speed camera, an experiment about the motion of microscale xed exible Journal of Nanotechnology ber was carried out to verify the long exible ber model.e xed ber length L f 100 mm, and the linear density (mass per unit length) ρ L 0.2 × 10 −5 kg•m −1 .As shown in Figure 5, the noncontact test bed consists of an air jet loom, high-speed camera, and professional image analysis software Image Pro-Plus.e ber is xed at the nozzle so its motion state can be recorded by the high-speed camera after each parameter adjustment of the loom.e transient ber shape comparison of numerical simulation results and that recorded by the high-speed camera are shown in Figure 6.It can be seen that the It can be seen that the free end of the ber uctuates significantly all the time, but the uctuation of the ber segment close to the main nozzle is not obvious.uctuates signicantly all the time, but the uctuation of the ber segment close to the main nozzle is not obvious.is is due to the turbulence characteristics of the jet ow, ber exibility, and the shape characteristics of the ber.
ere are some less signi cant di erences between the simulation models used in this paper and the actual ber properties: the bending modulus of the ber, which is much lower than the elasticity modulus, is not taken into account; the hydrodynamic forces such as Basset force and sliprotational lift force, which are relative small compared to the drag force, are neglected as well.Despite all the simpli cations, the experimental and simulation results are still in relatively good agreement (Figure 6).erefore, the following simulation of the ight ber is reasonable and credible.

Discussions and Analysis.
When a micro-or nanoscale free ber is being carried by the high-velocity air ow after being injected into the jet ow eld, as described above, the exible ber model assumes that the ber mass is concentrated at a series of beads, and the intervals between every two beads can vary as a result of the force-displacement balance on the beads.Subsequently, the transient equivalent radius of the ber varies too; they can even be reduced from microscale to nanoscale.Fiber linear density is another important characteristic and has a signi cant e ect on calculating the force acted on the beads which would determine the ber motion in the jet ow.So the ber motion with di erent linear densities of ρ L 0.5 × 10 −5 , 1.0 × 10 −5 , 1.5 × 10 −5 , and 2.0 × 10 −5 kg•m −1 are separately studied.
e initial velocity of the ber is ν in 70 m/s.e initial length of static free ber is L f 40 mm.e initial bead radius is given as R 100 μm.e two ends of the bers are marked as end A and B, respectively.At the initial stage, end A is at the position x 40 mm and end B is at x 0 mm (Figure 7).e motional characteristic of the ber is stable at the initial stage.e farther from the nozzle, the lower the air velocity.So, the velocity of end B will be faster than that of end A after a period of time.As a result, ber bending deformation is observed.

E ect of Fiber Linear Density on Fiber Fluctuation.
Due to di erent densities, the time of ber ying across the simulation domain is di erent.e ber movement is analyzed until the end A of ber arrives at the calculating boundary.
e motion and bending deformation of bers with di erent linear densities over time are shown in Figure 8.As the jet ow goes on, the vortex keeps rolling-up, transporting  6 Journal of Nanotechnology and mixing with each other (Figure 1), and the uctuation of vortex driven on ber becomes more and more strong.By comparing Figures 8(a)-8(d), it can be seen that the ber uctuation relating with linear density ρ L 0.5 × 10 −5 kg•m −1 is more obvious than that of ber with linear density ρ L 2.0 × 10 −5 kg•m −1 , which shows that the smaller linear density causes the more unstable ber motion.is is because the ber transverse acceleration generates from the velocity di erence between transverse velocity of ow eld and that of ber.Moreover, the smaller linear density results in the larger transverse acceleration and the larger transverse acceleration lead to the more unstable ber motion.
e transverse velocity of end A of bers with di erent linear densities is shown in Figure 9. e ber transverse velocity is 0 m/s at the initial time, and it becomes larger with the increasing of running time.e end transverse velocity of ber with linear density ρ L 2.0 × 10 −5 kg•m −1 increases to about 2 m/s after 0.0012 s.At the same time, the end transverse velocity of ber with linear density ρ L 0.5 × 10 −5 kg•m −1 increases to about 18 m/s.us, the smaller ber linear density leads to the larger transverse velocity.In other words, the ber transverse uctuation becomes more obvious.
In order to learn more about the in uence of air velocity on ber uctuations, several equidistant points are selected on the ber (Figure 10).To represent the amplitude of ber uctuation, the uctuation standard deviation of ber is calculated using (7) when the end A of ber arrives at 40 mm, 80 mm, 100 mm, 120 mm, and 140 mm (i.e., x A 40 mm, 80 mm, 100 mm, 120 mm, and 140 mm) from the main nozzle: where Δy is the vertical distance from the point to the center line and n is the sum of the selected points in the ber (n 41).e uctuation standard deviation is listed in Table 1.
As shown in Table 1, the ber is a ected by the air ow and the ber uctuation standard deviation increases gradually in the motion process with time.When the ber with di erent linear densities arrives at the same location in jet ow, the smaller ber linear density causes the greater uctuation standard deviation.e uctuation amplitude of bers with ρ L 0.5 × 10 −5 kg•m −1 is two times larger than that of bers with ρ L 2.0 × 10 −5 kg•m −1 .at is to say, the ber with smaller linear density will uctuate more obviously as it ies across the computation domain.As the cross section of the reed groove in the loom is about 5 mm × 5 mm, so when the uctuating amplitude of the ber end is larger than 2.5 mm, that is, after the head of ber with ρ L 0.5 × 10 −5 kg•m −1 arrived at x A 120 mm and the head of ber with ρ L 1.0 × 10 −5 kg•m −1 arrived at x A 140 mm, the ber will much more likely impact with the groove if there is no another assistant jet ow.So it is strongly recommended that the distance of the rst assistant nozzle from the main nozzle is less than 120 mm for the ber whose linear density is less than 0.5 With the increase of ber linear density, the distance could be appropriately increased.

E ect of Fiber Linear Density on Fiber Velocity.
e x-velocity (v A ) and distance from the nozzle (x A ) of end A with di erent linear densities and time are shown in Figure 11.
e corresponding ow axial velocity (at the positions where the end A is) is given in Figure 12.Within the running time of 0-0.0002 s, the ber velocity increases obviously.en, the ber is in a state of slow acceleration during the time of 0.0002-0.0004s.At the next stage, there is no longer a signi cant change on ber velocity.After about 0.0008 s, the ber is in a state of deceleration.It can be explained as follows.
e ber velocity is relatively low at the initial moment.When the end A velocity is around 70 m/s, the air velocity of ow eld is around 160 m/s (Figure 12).ere is a large velocity di erence between the ow eld and ber.e ber is in rapid acceleration because of the large drag force as listed previously.It means the ber gets a lot of energy from the ow in the period of 0-0.0002 s, and the end A ies about 15 mm forward in this ultra-short time.en, the velocity of ow eld, where the ber is, decreases rapidly, which we can see from Figure 12.During the time of Journal of Nanotechnology 0.0002-0.0005s, the velocity of air ow is larger than the ber, but the velocity di erence decreases.So the ber is in a state of slow acceleration.As the velocity di erence between the ow eld and ber keeps decreasing in 0.0005-0.0008s, the in uence of ow eld on ber velocity becomes unapparent and the ber remains at the same velocity magnitude especially for the bers with a relatively large linear density.At t 0.0008 s, the ber end A arrives around 100 mm away from the main nozzle, which we can see from Figure 11, and the velocity of air ow is lower than that of the ber.e ber cannot get energy from the ow any more for all the bers simulated here.In other words, the e cient length of the main nozzle to carry the ber ying rapidly forward is about 100-110 mm.So the addition of an assistant nozzle is suggested in this place.
Besides, the di erence of velocity distribution is obvious with di erent ber linear densities.According to Newton's second law, the lower linear density leads to greater ber acceleration.So when the velocity of air ow is greater than that of the ber, the greater ber acceleration causes the greater velocity.But, when the velocity of air ow is lower than that of the ber, the velocity of ber with a lower density falls quickly.It means that the ber with lower linear density is more sensitive to the ow, and the ow velocity change along the ber will easily cause a nonuniform velocity distribution at di erent segments of the ber.e x-velocity of ends A and B of the ber with di erent linear densities is shown in Figure 13.e velocity di erence between end A and B of the ber with liner density ρ L 0.5 × 10 −5 kg•m −1 is about 35 m/s when the end A arrives at the calculating boundary, and the ber with linear density ρ L 2.0 × 10 −5 kg•m −1 is only about 10 m/s.e nonuniform velocity distribution leads to obvious bending deformation of the ber (Figure 8(a)), which is undesirable for pneumatic conveying especially jet loom.In order to guarantee the stability of movement of the ber with lower   density, a more stable velocity distribution of flow is needed.On the other hand, the decrease of the distance between first the assistant nozzle and the main nozzle can reduce the axial velocity change and improve the stability of flow [28].So, in order to make the flight more stable, for the fiber with lower linear density, the distance of first assistant nozzle from the main nozzle should be appropriately smaller.But, it is obviously worse for saving energy.Actually, in industrial production, people will try their best to add the spacing to make full use of the high-speed air jet flow.
e discussions above are summarized as follows.First, as we have discussed before, for fiber with linear density ρ L � 0.5 × 10 −5 kg•m −1 , if there is an assistant nozzle within 120 mm from the main nozzle, it will be highly likely for the fiber to impact with the groove.Second, the efficient length of the main nozzle to carry the fiber forward is less than 110 mm.ird, to make full use of the high-speed air jet flow, the distance between the main nozzle and the first assistant nozzle cannot be very small, although the smaller nozzle spacing could lead to a more stable fiber flight.Considering all the factors, there is only a narrow range of suggestion distance to set the first assistant nozzle.For fiber with a linear density ρ L � 0.5 × 10 −5 kg•m −1 , it is 110 mm, and when the fiber linear density increases, the distance could be appropriately increased to 140 mm.

Conclusions
To make the suggestion of how to make the fiber fly with high efficiency and stability in a jet flow, we employed the LES method to simulate the development of vortices in the jet flow and Lagrangian bead-rod model to give the time evolution of long flexible fiber distribution with different linear densities.e fluctuation and velocity of fiber in jet flow were then studied numerically, and the simulation results can provide an optimization option for the air-jet loom to improve the energy efficiency by reasonably arranging the first assistant nozzle.e results are as follows.
(1) As the primary vortex rolls up, transports, and mixes with each other, the fiber fluctuation becomes stronger and the motion becomes more unstable as the linear density decreases.(2) e fluctuation amplitude of a fiber with ρ L � 0.5 × 10 −5 kg•m −1 is two times larger than that of a fiber with ρ L � 2.0 × 10 −5 kg•m −1 .It is as large as 3.34 mm when the fiber with ρ L � 0.5 × 10 −5 kg•m −1 arrives at x A � 120 mm.(3) e distance of the first assistant nozzle from the main nozzle should be less than 120 mm to avoid the collision between the fiber and the loom.With the increasing fiber linear density, the distance could be appropriately increased.(4) e efficient length of the main nozzle to carry the fiber flying steadily forward is about 100-110 mm.So an assistant nozzle should be added in this place.(5) To save energy, according to (2) and (3), the suggested distance between the main nozzle and the first assistant nozzle is 110 mm for the fiber with ρ L � 0.5 × 10 −5 kg•m −1 .When the fiber linear density increases, the distance could be appropriately increased to 140 mm.

Figure 1 :
Figure 1: Schematic diagram of vortices in jet ow.

Figure 2 :
Figure 2: Schematic diagram of the exible ber model.

5 Figure 4 :
Figure 4: Self-preservation pro les of the mean streamwise velocity by the LES method.

Figure 9 :Figure 10 :
Figure 9: Transverse velocity distribution of end A of bers with di erent linear densities.

Figure 12 :Figure 13 :
Figure 12: Air ow velocity at the position of ber end A.

Figure 11 :
Figure 11: v A and x A curves with di erent linear densities and time.

Table 1 :
Standard deviation of ber uctuations with di erent ber linear densities at di erent positions.