Research ArticleSimulation of Motion of Long Flexible Fibers with DifferentLinear Densities in Jet Flow

Peifeng Lin ,1 Wenqian Xu,1 Yuzhen Jin,1 and Zefei Zhu 2

1Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310008, China2School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310008, China

Correspondence should be addressed to Zefei Zhu; zzf.3691@163.com

Received 1 February 2018; Accepted 12 March 2018; Published 2 May 2018

Academic Editor: Mingzhou Yu

Copyright © 2018 Peifeng Lin et al. ,is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Air-jet loom is a textile machine designed to drive the long fiber using a combination flow of high-pressure air from a main nozzleand a series of assistant nozzles. To make the suggestion of how to make the fiber fly with high efficiency and stability in the jetflow, in which vortices also have great influence on fiber movement, the large eddy simulation method was employed to obtain thetransient flow field of turbulent jet, and a bead-rod chain fibermodel was used to predict long flexible fiber motion.,e fluctuationand velocity of fibers with different linear densities in jet flow were studied numerically. ,e results show that the fluctuationamplitude of a fiber with a linear density of 0.5×10−5 kg·m−1 is two times larger than that of a fiber with a linear density of2.0×10−5 kg·m−1. ,e distance of the first assistant nozzle from the main nozzle should be less than 120mm to avoid collisionbetween the fiber and the loom. ,e efficient length of the main nozzle to carry the fiber flying steadily forward is about100–110mm. For fibers with a linear density of 0.5×10−5 kg·m−1, it is suggested that the distance of the first assistant nozzle fromthe main nozzle is about 110mm. With the increase of fiber linear density, the distance could be appropriately increased to140mm.,e simulation results provide an optimization option for the air-jet loom to improve the energy efficiency by reasonablyarranging the first assistant nozzle.

1. Introduction

Carried flight of a long, flexible fiber in a jet flow field isa fundamental science problem for air-jet loom [1], which isdesigned to drive the yarn using a combination flow of high-pressure air from a main nozzle and a series of assistantnozzles. Different from other pipeline pneumatic conveyingsystems, the jet flow of the main nozzle will rapidly de-celerate in a free area and cannot guarantee a sustained high-speed flight of the fiber. ,erefore, assistant nozzles must beadded to the system. On the other hand, the presence of thereed groove requires that the fluctuation of the fiber must beless than a certain range. How to make the fiber fly forwardrapidly and steadily is always a core issue of fiber flight and isjointly determined by two aspects: the fiber model and theflow simulation.

In general, a long fiber can bemodeled as a rigid or flexiblefiber for different dynamic research purposes. Yamamoto and

Matsuoka [2] proposed a method for simulating the dynamicbehavior of rigid and flexible fibers in a flow field. ,e fiber isregarded asmade up of spheres that are lined up and bound toeach neighbor. Each pair of bonded spheres can stretch, bend,and twist by changing the bond distance, bond angle, andtorsion angle between spheres, respectively. For the rigidfiber, the computed period of rotation and distribution oforientation angle agree with those calculated using Jeffery’sequation with an equivalent ellipsoidal aspect ratio. For theflexible fiber, the period of rotation decreases rapidly with thegrowth of bending deformation of the fiber and rotation orbitsdeviate from a circular one for the rigid fiber. De Meule-meester et al. [3] tried to study the dynamic properties of fibermovement and developed a one-dimensional mathematicalmodel, in which the behavior of the fiber is described byNewton’s second law. Lindstrom and Uesaka [4] proposeda model for flexible fibers in viscous fluid flow; namely, thefibers are modeled as chains of fiber segments which interact

HindawiJournal of NanotechnologyVolume 2018, Article ID 8672106, 10 pageshttps://doi.org/10.1155/2018/8672106

with the �uid through viscous and dynamic drag forces.Fiber segments, from the same or di�erent �bers, interactwith each other through normal, frictional, and lubricationforces. �e simulations using the proposed model suc-cessfully reproduced the di�erent regimes of motion forthread-like particles that range from rigid �ber motionto complicated orbiting behavior including coiling andself-entanglement.

Vahidkhah and Abdollahi [5] used the lattice Boltzmannmethod (LBM) to solve the Newtonian �ow �eld and theimmersed boundary method (IBM) to simulate the de-formation of the �exible �ber interacting with the �ow. �evariations of the �ber length during the simulation time fordi�erent values of stretching constant are studied. Kabanemiand Hetu [6] carried out a direct simulation study to analyzethe e�ect of �ber rigidity on �ber motion in simple shear �ow.�e �ber is modeled as a series of rigid spheres connected bysti� springs, which is similar to that used by Yamamoto andMatsuoka [2].�emodel 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 insimple shear �ow. Meirson and Hrymak [7] extracted rota-tional friction coe�cients from Je�ery’s model, created ageneral case long �exible �ber orientation model, and appliedit in a simple shear �ow. Nan et al. [8] presented a linearviscoelastic sphere-chain model based on the discrete elementmethod to quantify thematerial damping of deformed �exible�bers. A correlation is formulated to quantify the relationshipbetween the damping coe�cient of the local bond and that ofthe �exible �ber. Meulemeester et al. [9] developed a three-dimensional mathematical model of the yarn. �e three-dimensional model for the weft insertion on air-jet loomshas been successfully tested.

On the other side, as we have mentioned, �ber �ying in�uid is also a�ected by turbulent �ow characteristics. Kimet al. [10] analyzed the �ow in an air-jet loom by usinga time-accurate characteristic-based upwind �ux-di�erencesplitting compressible Navier–Stokes method. �e unsteadypressure and Mach number behavior along the center line ofthe main nozzle were analyzed. Andric et al. [11] analyzed thedynamics of individual �exible �bers in a turbulent �ow, thedirect numerical simulation of the incompressible Navier–Stokes equations is used to describe the �uid �ow in a planechannel, and a one-way coupling is considered between the�bers and the �uid phase. �ey found that the �ber motion isprimarily governed by velocity correlations of the �ow �uc-tuations. In addition, they reported that there is a clear ten-dency of the thread-like �bers to evolve into complexgeometrical con�gurations in a turbulent �ow �eld, and the�ber inertia has a signi�cant impact on reorientation time-scales of �bers suspended in a turbulent �ow �eld. Jin et al. [12]simulated the turbulent �ow by solving the Reynolds-averaged Navier–Stokes (RANS) equations and con-ducted the three-dimensional numerical simulation of themovement of the �exible body. �e numerical resultsshow that an unconstrained �exible body would turn overforward along the air�ow’s di�usion direction, whilea constrained �exible body in the �ow �eld will makea periodic rotation motion along the axis of the �exible

body, and the bending deformation is more obvious than thatof unconstrained �exible body. Yang et al. [13] studied thetwo-way coupling turbulent model and rheological propertiesfor �ber suspension in the contraction based on the RANSsimulation.

Pei and Yu [14] studied the motional characteristics ofthe �exible �bers in the air�ow inside the Murata vortexspinning (MVS) nozzle. A two-dimensional �uid structureinteraction (FSI) model combined with the �ber wall contactis introduced to simulate a single �ber moving in the air�owinside the MVS nozzle. �e model is solved using a �niteelement code ADINA. Based on their simulation results, theformation principle and the in�uence of some nozzle pa-rameters on the tensile property of the MVS �ber werediscussed.

More and more investigators put their interest on theair-jet loom [15–18] and other �uid machineries [19, 20],but due to the computation cost gap between scienti�cresearches and engineering needs, only a few of themconsidered the in�uence of turbulent �uctuation [21]. �ejet �ow caused by the main nozzle is a typical free shearturbulence, and there are strong vortices which playan important role on the momentum and energy transportof �ow �eld [22, 23]. �e vortices also have great in�uenceon the �ber movement. Large eddy simulation (LES) isa believed mathematical model for turbulence vortexsimulation, by directly calculating the large-scale �owmotion.

In this paper, we simpli�ed the two-phase �ow systemand were able to employ the LES method to simulate thedevelopment of a vortex in jet �ow and the Lagrangian bead-rod model to give the time evolution of a long �exible �berdistribution with di�erent linear densities. �en, the �ber�uctuation and the velocity were discussed to make thesuggestion of how to make the �ber �y with high e�ciencyand stability in an air-jet loom.

2. Mathematical Formulation

2.1. FluidFlow. �e 2D jet �ow is shown in Figure 1, in whichx and y are the streamwise and cross-stream directions, re-spectively. �e width of the nozzle D is 3.5mm, the �uidvelocity at the nozzle U0 is 240m/s, and the �ow Reynoldsnumber Re � 5.68 × 104.

y

x

D

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

2 Journal of Nanotechnology

�e LES equations governing the jet �ow obtained by�ltering the Navier–Stokes equations are as follows [24]:

zρzt+z ρun( )zxn

� 0,

z ρum( )zt

+z ρumun( )

zxn�−zpzxm

+z

zxnμzσmnzxn− τmn( ),

(1)

where ρ is the �uid density, um is the �ltered velocity, p is the�ltered pressure, index m, n is taken as 1, 2 and refers to thex, y, and μ is the dynamic viscosity.

�e SGS (subgrid stress) tensor τmn and Smargorinsky–Lilly model [24] which are based on the mixing length hy-pothesis are used to calculate the SGS stress.

2.2. Flexible Fiber Model. A single �exible long �ber ismodeled as a bead-rod chain, which is similar to that used byGuo et al. [25]. �e �ber model is composed of N beads,which are connected byN− 1 massless rods (Figure 2). Onlythe beads are a�ected by forces, and the rods maintain thecon�guration of the �ber. Using the model, the chain isallowed to be stretched by changing the distance of adjacentrods.

If the distance between adjacent beads is not equal to theequilibrium distance, the stretching restoring force Fniexerted on the bead i will be

Fni � −πR2E

LΔL, (2)

where R is the bead radius, E is the elastic modulus, L is theequilibrium distance, and ΔL is the distance variation, whichis equal to the transient distance of each two adjacent beadssubtracted by the equilibrium distance.

When immersed in the unsteady �ow �eld which iscalculated in the last section, the �ber is subjected to hy-drodynamic forces, which are also changed with time. In thispaper, only drag force is considered in order to decrease thecomputational cost. Other hydrodynamic forces such asBasset history term, additional mass, slip-rotational liftforce, and �uid inertia are negligible. For bead i, the dragforce Fdi is contributed by �ber sections (i− 1, i) and(i, i + 1). It can be calculated as follows:

Fdi �12

Fidi−1, i + Fidi, i+1( ), (3)

where Fdi−1, i and Fdi, 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 Fdi−1, i acting on the �ber section(i− 1, i) can be expressed as follows:

Fidi−1,i �πr2v2Cdρ Vqi −Vdi

∣∣∣∣∣∣∣∣∣∣ Vqi −Vdi( ), (4)

where Vqi and Vdi are the �uid and �ber velocities at themass center of the �ber section (i− 1, i), and Cd is the dragcoe�cient associated with Reynolds number, which can berepresented as follows:

Cd �

24Re

Re≤ 1,

24Re+

4���Re3

√ 1<Re≤ 1000,

β Re> 1000,

(5)

where Res � 2rvρ|Vqi −V0i| is the Reynolds number ofequivalent spherical and β is a constant between 0.4 and 0.45.According to Newton’s second law, the equations of motionfor the bead i that constitute the �ber are as follows:

mid2ridt2

� F,

F � Fni + Fdi,

(6)

wheremi is the mass of the bead i and ri is the position vectorof the bead i.

3. Results and Discussion

3.1. Computation Conditions. �e computational domaincovers x × y � 21D × 40D, the nozzle width D� 3.5mm,fully developed boundary conditions at the outlet, and staticsurrounding environment conditions are assumed.�e localmesh is shown in Figure 3. Fiber motion is solved in one-waycoupling between the �ber and �ow; �ber-wall and �ber-�ber interactions are neglected.

3.2. Method Veri�cation

3.2.1. Jet Flow Field Veri�cation. According to the experi-mental results of [26, 27], the streamwise velocity satis�esthe self-preservation distribution at x≥ 8D and the pro�le isexpressed as a Gaussian curve. �e self-preservation pro�lesof the mean velocity are shown in Figure 4, where the dataare normalized by the centerline mean velocity Um and thehalf-width r0.5, which is the distance between the positionwhere the streamwise velocity u/Um � 0.5 and the centerline of the jet. �e �gure shows that the air �ow which ismentioned in Section 2.1 has been successfully simulated.

3.2.2. Flexible Fiber Model Veri�cation. As the limitation ofthe visible area and the resolution of the high-speed camera,an experiment about the motion of microscale �xed �exible

1

N

i i + 1

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

Journal of Nanotechnology 3

�ber was carried out to verify the long �exible �ber model.�e �xed �ber length Lf � 100mm, and the linear density(mass per unit length) ρL � 0.2 × 10−5 kg·m−1. As shown inFigure 5, the noncontact test bed consists of an air jet loom,high-speed camera, and professional image analysis softwareImage Pro-Plus. �e �ber is �xed at the nozzle so its motionstate can be recorded by the high-speed camera after eachparameter adjustment of the loom.

�e transient �ber shape comparison of numericalsimulation results and that recorded by the high-speedcamera are shown in Figure 6. It can be seen that the Itcan be seen that the free end of the �ber �uctuates signif-icantly all the time, but the �uctuation of the �ber segmentclose to the main nozzle is not obvious. �uctuates signi�-cantly all the time, but the �uctuation of the �ber segmentclose to the main nozzle is not obvious. �is is due to theturbulence characteristics of the jet �ow, �ber �exibility, andthe shape characteristics of the �ber.

�ere are some less signi�cant di�erences between thesimulation models used in this paper and the actual �berproperties: the bending modulus of the �ber, which is muchlower than the elasticity modulus, is not taken into account;the hydrodynamic forces such as Basset force and slip-rotational lift force, which are relative small compared tothe drag force, are neglected as well. Despite all the sim-pli�cations, the experimental and simulation results are stillin relatively good agreement (Figure 6). �erefore, thefollowing simulation of the �ight �ber is reasonable andcredible.

3.3. Discussions and Analysis. When a micro- or nanoscalefree �ber is being carried by the high-velocity air�ow afterbeing injected into the jet �ow �eld, as described above, the�exible �ber model assumes that the �ber mass is con-centrated at a series of beads, and the intervals between everytwo beads can vary as a result of the force-displacementbalance on the beads. Subsequently, the transient equivalentradius of the �ber varies too; they can even be reduced frommicroscale to nanoscale. Fiber linear density is anotherimportant characteristic and has a signi�cant e�ect oncalculating the force acted on the beads which would de-termine the �ber motion in the jet �ow. So the �ber motionwith di�erent linear densities of ρL � 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 νin � 70 m/s. �einitial length of static free �ber is Lf� 40mm.�e initial beadradius is given as R� 100 μm. �e two ends of the �bers aremarked as end A and B, respectively. At the initial stage, endA is at the position x� 40mm and end B is at x� 0mm(Figure 7).�emotional characteristic of the �ber is stable atthe initial stage.�e farther from the nozzle, the lower the airvelocity. So, the velocity of end B will be faster than that ofend A after a period of time. As a result, �ber bendingdeformation is observed.

3.3.1. E�ect of Fiber Linear Density on Fiber Fluctuation. Dueto di�erent densities, the time of �ber �ying across thesimulation domain is di�erent. �e �ber movement is an-alyzed until the end A of �ber arrives at the calculatingboundary.

�e motion and bending deformation of �bers withdi�erent linear densities over time are shown in Figure 8. Asthe jet �ow goes on, the vortex keeps rolling-up, transporting

8D12D

16DGaussian

0.0

0.5

1.0

u/U

m

–2 –1 0 1 2 3–3y/r0.5

Figure 4: Self-preservation pro�les of the mean streamwise ve-locity by the LES method.

Figure 5: Test equipment of the experiment.

0.002 0.004 0.006 0.008 0.01 0.0120x

–0.006

–0.004

–0.002

0

0.002

0.004

0.006

y

Figure 3: Mesh at the nozzle exit.

4 Journal of Nanotechnology

–0.03

–0.02

–0.01

0.00

0.01

0.02

0.03Po

sitio

n y (

m)

0.02 0.04 0.06 0.08 0.100.00Position x (m)

0.0030s

(a)

–0.03

–0.02

–0.01

0.00

0.01

0.02

0.03

Posit

ion y (

m)

0.02 0.04 0.06 0.08 0.100.00Position x (m)

0.0040s

(b)

–0.03

–0.02

–0.01

0.00

0.01

0.02

0.03

Posit

ion y (

m)

0.02 0.04 0.06 0.08 0.100.00Position x (m)

0.0050s

(c)

–0.03

–0.02

–0.01

0.00

0.01

0.02

0.03

Posit

ion y (

m)

0.02 0.04 0.06 0.08 0.100.00Position x (m)

0.0055s

(d)

Figure 6: Result comparison of �ber between numerical simulation and high-speed photography.

Journal of Nanotechnology 5

AB B BAA

–10

–5

0

5

10

y/D

5 10 15 20 25 30 35 400x/D

t = 0st = 0.000650st = 0.001200s

Figure 7: Motion of free �ber in jet �ow.

–10

–5

0

5

10

y/D

5 10 15 20 25 30 35 400x/D

t = 0st = 0.000625st = 0.001100s

(a)

–10

–5

0

5

10y/D

5 10 15 20 25 30 35 400x/D

t = 0st = 0.000650st = 0.001200s

(b)

–10

–5

0

5

10

y/D

5 10 15 20 25 30 35 400x/D

t = 0st = 0.000675st = 0.001250s

(c)

–10

–5

0

5

10

y/D

5 10 15 20 25 30 35 400x/D

t = 0st = 0.000675st = 0.001275s

(d)

Figure 8: Motional characteristic of the �ber with di�erent linear densities: (a) ρL � 0.5 × 10−5 kg·m−1, (b) ρL � 1.0 × 10−5 kg·m−1,(c) ρL � 1.5 × 10−5 kg·m−1, and (d) ρL � 2.0 × 10−5 kg·m−1.

6 Journal of Nanotechnology

and mixing with each other (Figure 1), and the �uctuationof 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 ρL � 0.5 × 10−5 kg·m−1is more obvious than that of �ber with linear densityρL � 2.0 × 10−5 kg·m−1, which shows that the smaller lineardensity causes the more unstable �ber motion. �is is be-cause the �ber transverse acceleration generates from thevelocity di�erence between transverse velocity of �ow �eldand that of �ber. Moreover, the smaller linear density resultsin the larger transverse acceleration and the larger transverseacceleration lead to the more unstable �ber motion.

�e transverse velocity of end A of �bers with di�erentlinear densities is shown in Figure 9. �e �ber transversevelocity is 0m/s at the initial time, and it becomes larger withthe increasing of running time. �e end transverse velocityof �ber with linear density ρL � 2.0 × 10−5 kg·m−1 increasesto about 2m/s after 0.0012 s. At the same time, the endtransverse velocity of �ber with linear density ρL � 0.5 ×10−5 kg·m−1 increases to about 18m/s.�us, the smaller �berlinear density leads to the larger transverse velocity. In otherwords, the �ber transverse �uctuation becomes moreobvious.

In order to learn more about the in�uence of air velocityon �ber �uctuations, several equidistant points are selectedon the �ber (Figure 10). To represent the amplitude of �ber�uctuation, the �uctuation standard deviation of �ber iscalculated using (7) when the end A of �ber arrives at 40mm,80mm, 100mm, 120mm, and 140mm (i.e., xA � 40mm,80mm, 100mm, 120mm, and 140mm) from the mainnozzle:

σ �

������∑Δy2

n

√

, (7)

where Δy is the vertical distance from the point to thecenter line and n is the sum of the selected points in the�ber (n� 41). �e �uctuation standard deviation is listed inTable 1.

As shown in Table 1, the �ber is a�ected by the air�owand the �ber �uctuation standard deviation increasesgradually in the motion process with time. When the �berwith di�erent linear densities arrives at the same location injet �ow, the smaller �ber linear density causes the greater�uctuation standard deviation. �e �uctuation amplitudeof �bers with ρL � 0.5 × 10−5 kg·m−1 is two times larger thanthat of �bers with ρL � 2.0 × 10−5 kg·m−1. �at is to say, the�ber with smaller linear density will �uctuate more obvi-ously as it �ies across the computation domain. As the crosssection of the reed groove in the loom is about5mm × 5mm, so when the �uctuating amplitude of the�ber end is larger than 2.5mm, that is, after the head of�ber with ρL � 0.5 × 10−5 kg·m−1 arrived at xA � 120mmand the head of �ber with ρL � 1.0 × 10−5 kg·m−1 arrived atxA � 140mm, the �ber will much more likely impact withthe groove if there is no another assistant jet �ow. So it isstrongly recommended that the distance of the �rst as-sistant nozzle from the main nozzle is less than 120mm forthe �ber whose linear density is less than 0.5×10−5 kg·m−1.

With the increase of �ber linear density, the distance couldbe appropriately increased.

3.3.2. E�ect of Fiber Linear Density on Fiber Velocity. �ex-velocity (vA) and distance from the nozzle (xA) of end A withdi�erent linear densities and time are shown in Figure 11.�e corresponding �ow axial velocity (at the positions wherethe end A is) is given in Figure 12. Within the running timeof 0–0.0002 s, the �ber velocity increases obviously. �en,the �ber is in a state of slow acceleration during the time of0.0002–0.0004 s. At the next stage, there is no longer a sig-ni�cant change on �ber velocity. After about 0.0008 s, the�ber is in a state of deceleration. It can be explained asfollows. �e �ber velocity is relatively low at the initialmoment. When the end A velocity is around 70m/s, the airvelocity of �ow �eld is around 160m/s (Figure 12). �ere isa large velocity di�erence between the �ow �eld and �ber.�e �ber is in rapid acceleration because of the large dragforce as listed previously. It means the �ber gets a lot ofenergy from the �ow in the period of 0–0.0002 s, and the endA �ies about 15mm 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

–5

0

5

10

15

20

Velo

city

v y (m

/s)

0.00050 0.000750.00025 0.00100 0.001250.00000Time t (s)

0.5e – 5 kg/m1.0e – 5 kg/m

1.5e – 5 kg/m2.0e – 5 kg/m

Figure 9: Transverse velocity distribution of end A of �bers withdi�erent linear densities.

B

A

Δy

xA

Figure 10: Illustration of free �ber in jet �ow.

Journal of Nanotechnology 7

0.0002–0.0005 s, the velocity of air�ow is larger than the �ber,but the velocity di�erence decreases. So the �ber is in a stateof slow acceleration. As the velocity di�erence between the�ow �eld and �ber keeps decreasing in 0.0005–0.0008 s,the in�uence of �ow �eld on �ber velocity becomes un-apparent and the �ber remains at the same velocity mag-nitude especially for the �bers with a relatively large lineardensity. At t� 0.0008 s, the �ber end A arrives around100mm 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 forall the �bers simulated here. In other words, the e�cientlength of the main nozzle to carry the �ber �ying rapidlyforward is about 100–110mm. So the addition of an assistantnozzle is suggested in this place.

Besides, the di�erence of velocity distribution is obviouswith di�erent �ber linear densities. According to Newton’ssecond law, the lower linear density leads to greater �beracceleration. So when the velocity of air�ow is greater thanthat of the �ber, the greater �ber acceleration causes thegreater velocity. But, when the velocity of air�ow is lowerthan that of the �ber, the velocity of �ber with a lowerdensity falls quickly. It means that the �ber with lower lineardensity is more sensitive to the �ow, and the �ow velocitychange along the �ber will easily cause a nonuniform ve-locity distribution at di�erent segments of the �ber.

�e x-velocity of ends A and B of the �ber with di�erentlinear densities is shown in Figure 13.�e velocity di�erencebetween end A and B of the �ber with liner density ρL �0.5 × 10−5 kg·m−1 is about 35m/s when the end A arrives atthe calculating boundary, and the �ber with linear densityρL � 2.0 × 10−5 kg·m−1 is only about 10m/s. �e non-uniform velocity distribution leads to obvious bendingdeformation of the �ber (Figure 8(a)), which is undesirablefor pneumatic conveying especially jet loom. In order toguarantee the stability of movement of the �ber with lower

0.00050.0000 0.00150.0010Time t (s)

–50

0

50

100

150

200

250

Velo

city

v f (m

/s)

0.5e – 5 kg/m1.0e – 5 kg/m

1.5e – 5 kg/m2.0e – 5 kg/m

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

0.00000 0.00025 0.00050Time t (s)

0.00075 0.00100 0.0012570

80

90

100

110

120

Velo

city

v (m

/s)

vA vB vA vB0.5e – 5 kg/m1.0e – 5 kg/m

1.5e – 5 kg/m2.0e – 5 kg/m

Figure 13: X-velocity of ends A and B of �ber with di�erent lineardensities.

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

Linear density Amplitude of �ber �uctuation, σ (mm)ρL (kg·m–1) xA� 40mm xA� 80mm xA� 100mm xA� 120mm xA� 140mm0.5×10−5 0 0.75 1.62 3.34 5.031.0×10−5 0 0.40 1.01 2.04 3.201.5×10−5 0 0.29 0.75 1.54 2.392.0×10−5 0 0.22 0.61 1.13 1.81

0.0005 0.0010 0.00150.0000Time t (s)

70

75

80

85

90

95

100

Velo

city

v A (m

/s)

405060708090100110120130140150

Posit

ion,

xA

(mm

)

vA xA vA xA0.5e – 5 kg/m1.0e – 5 kg/m1.5e – 5 kg/m

2.0e – 5 kg/mThe distance of v = 70e3 mm/s

Figure 11: vA and xA curves with di�erent linear densities and time.

8 Journal of Nanotechnology

density, a more stable velocity distribution of flow is needed.On the other hand, the decrease of the distance between firstthe assistant nozzle and the main nozzle can reduce the axialvelocity change and improve the stability of flow [28]. So, inorder to make the flight more stable, for the fiber with lowerlinear density, the distance of first assistant nozzle from themain nozzle should be appropriately smaller. But, it isobviously worse for saving energy. Actually, in industrialproduction, people will try their best to add the spacing tomake 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 densityρL � 0.5 × 10−5 kg·m−1, if there is an assistant nozzle within120mm from the main nozzle, it will be highly likely for thefiber to impact with the groove. Second, the efficient lengthof the main nozzle to carry the fiber forward is less than110mm. ,ird, to make full use of the high-speed air jetflow, the distance between the main nozzle and the firstassistant nozzle cannot be very small, although the smallernozzle spacing could lead to a more stable fiber flight.Considering all the factors, there is only a narrow range ofsuggestion distance to set the first assistant nozzle. For fiberwith a linear density ρL � 0.5 × 10−5 kg·m−1, it is 110mm,and when the fiber linear density increases, the distancecould be appropriately increased to 140mm.

4. Conclusions

To make the suggestion of how to make the fiber fly withhigh efficiency and stability in a jet flow, we employed theLES method to simulate the development of vortices in thejet flow and Lagrangian bead-rod model to give the timeevolution of long flexible fiber distribution with differentlinear densities. ,e fluctuation and velocity of fiber in jetflow were then studied numerically, and the simulationresults can provide an optimization option for the air-jetloom to improve the energy efficiency by reasonablyarranging the first assistant nozzle. ,e results are as follows.

(1) As the primary vortex rolls up, transports, and mixeswith each other, the fiber fluctuation becomesstronger and the motion becomes more unstable asthe linear density decreases.

(2) ,e fluctuation amplitude of a fiber with ρL � 0.5 ×

10−5 kg·m−1 is two times larger than that of a fiberwith ρL � 2.0 × 10−5 kg·m−1. It is as large as 3.34mmwhen the fiber with ρL � 0.5 × 10−5 kg·m−1 arrives atxA � 120mm.

(3) ,e distance of the first assistant nozzle from themain nozzle should be less than 120mm to avoid thecollision between the fiber and the loom. With theincreasing fiber linear density, the distance could beappropriately increased.

(4) ,e efficient length of the main nozzle to carry thefiber flying steadily forward is about 100–110mm. Soan assistant nozzle should be added in this place.

(5) To save energy, according to (2) and (3), the sug-gested distance between the main nozzle and the first

assistant nozzle is 110mm for the fiber with ρL �

0.5 × 10−5 kg·m−1. When the fiber linear density in-creases, the distance could be appropriately increasedto 140mm.

Conflicts of Interest

,e authors declare that they have no conflicts of interest.

Acknowledgments

,is work was supported by the National Natural ScienceFoundation of China (Grant nos. 51576180 and 51676173),Fluid Engineering Innovation Team of Zhejiang Sci-TechUniversity (ZSTU) of China (Grant no. 11132932611309),National Scholarship Foundation of China, and 521 TalentsFostering Program Funding of Zhejiang Sci-Tech Universityof China.

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