Effects of structural parameters on the tangentially injected swirling flow in concentric tubes with different lengths as a model of the vortex spinning nozzle

Abstract

The tangentially injected swirling tube flow has broad applications in various engineering fields. In the textile industry, the vortex spinning technology, which produces short-staple yarns by means of a tangentially injected swirling airflow, is of great prospect in view of its spinning speed and yarn structure. In this paper, the vortex spinning nozzle is abstracted into a model of concentric tubes of different lengths with tangential injectors. A computational fluid dynamics model is presented to study the effects of some structural parameters on the swirling flow generated in this tube system. The particle image velocimetry (PIV) technique is adopted to measure the flow field. Due to the difficulty in duplicating the Mach number, Reynolds numbers are matched between the tube model and prototype in the PIV experiment. Despite the discrepancy between the velocity values provided by the numerical simulation and experiment due to the compressibility effects not being reproduced in the PIV measurement, a comparative analysis concludes that qualitatively matched results on the flow pattern and effects of the structural parameters are obtained by the numerical simulation and PIV measurement.

Keywords

vortex spinning tangentially injected swirling flow concentric tubes parameter numerical simulation PIV

Tangentially injected swirling tube flow is generated by injecting a fluid into a cylindrical tube via injectors that are tangential to the tube wall. Since an inclination angle usually exists between the injector and the tube axis, the swirling flow generated in this way has tangential, radial, and axial velocity components, leading to wide and important applications in various industries. For example, in combustion systems, tangentially injected swirling flow is used for enhancing the mixing of the fuel and gases and to improve flame stabilization.¹ In heat exchangers, tangentially injected swirling flow has been adopted for enhancing heat transfer and slowing down the decay rate.² In the papermaking industry, separation of pulp fibers with different physical properties is achieved with the help of the centrifugal force generated in the hydrocyclone.³ Meanwhile, in the textiles industry, swirling flow has been used to directly insert twist into the yarn – after entering the cylindrical twisting tube, the fiber strand rotates around the tube axis under the action of the swirling airflow generated in the twisting tube through tangential injection. Twist is inserted into the fiber strand when it is moving downstream axially.⁴

Extremely high rotational rate of the yarn, up to perhaps 300,000 r/min,⁵ can be achieved by utilizing tangentially injected swirling tube flow in the yarn-spinning process. Among the modern spinning technologies using swirling airflow, vortex spinning is considered to be of the greatest potential, with the broadest prospects for its application due to its promising features of very high yarn delivery speed (up to 500 m/min), excellent spinnability, and fascinated yarn structure.^6,7 These features are closely related to the structure of its twisting tube, also known as the nozzle. Different from its predecessor – air-jet spinning, which incorporates two cylindrical nozzles in tandem – vortex spinning contains only one nozzle with tangential injectors but is equipped with a hollow spindle whose leading portion is inserted into the outlet section of the nozzle,^8–10 as shown in Figure 1. The hollow spindle is provided with a coaxial channel serving as the yarn delivery passage. Swirling flow is generated inside the twisting system after the injection of compressed air through the tangential injectors. Yarn properties are closely related to the airflow patterns and fiber dynamics in the nozzle. Although the fiber dynamics during the vortex spinning process have been studied by some researchers,^9,11–13 studies on the flow pattern in the nozzle still need to deepened and refined in order to better understand the physics behind it and provide a reference for the design of the nozzle, both theoretically and experimentally. In our previous studies,^14,15 the twisting system of vortex spinning was abstracted into a model of concentric tubes of different lengths with tangential injectors, in which a longer outer tube with tangential injectors represents the nozzle while a shorter inner tube represents the hollow spindle. In addition, the inlet of the inner tube lies in the vicinity of the injector exits, and the outlets of both tubes are set to lie on the same plane. The three-dimensional flow structure inside this system of concentric tubes was simulated and then analyzed using computational fluid dynamics (CFD), which has been a useful tool for studying the complex flow field of textile machinery.^16–18 An abstract of the nozzle geometry has been adopted because this work is not only dedicated to the specific problem at hand, but also to providing more general information on the behavior of the tangentially injected swirling tube flow that is employed in a large number of textile manufacturing processes such as air-jet spinning,¹⁷ Jetring spinning,¹⁹ air suction gun,²⁰ etc. In this work, we go a step further to investigate the effects of some tube structure parameters on the tangentially injected swirling flow inside the concentric tubes with different lengths using CFD. In addition, the particle image velocimetry (PIV) technique, which has been reported to be an effective measurement method for tube and pipe flows,^21,22 is adopted in the experiment to measure the swirling flow field in this tube system. Comparison is made and an analysis is presented on the differences between the numerical and experimental results. Flow fields employed in the textile manufacturing process using fluids are usually difficult to measure experimentally for verifying the simulation results owing to the high speed of the flow, small size of the components, and internal flow characteristics. Therefore, this work is not only aimed at providing a basis for further optimizing and designing a vortex spinning system, but is also expected to provide an effective and systematic way of solving the fluids engineering problems in the textile manufacturing process.

Figure 1.

(a) The structure of the twisting system of the Murata vortex spinning nozzle;¹⁰ (b) the Murata vortex spinning nozzle abstract model.

Numerical model

Governing equations and turbulence model

The tangentially injected swirling flow in the concentric tubes is considered to be compressible, turbulent Newtonian flow in three dimensions. In this research, only steady-state results are considered. Therefore, the Favre-averaged forms of the mean flow equations including the mass conservation equation, momentum equation, and energy equation in a Cartesian co-ordinate system are employed

div (\bar{ρ} \tilde{U}) = 0

(1)

div (\bar{ρ} \tilde{U} \tilde{U}) = - grad \bar{P} + div (μ grad \tilde{U} - \bar{\bar{ρ} u' u'}) + S_{M}

(2)

div (\bar{ρ} \tilde{T} \tilde{U}) = div (k grad \tilde{T} - \bar{\bar{ρ} u' T'}) + S_{T}

(3)

The following equation of state is used to relate the state variables ρ and T

\bar{P} = \bar{ρ} R \tilde{T}

(4)

where the overbar ^- indicates a Reynolds-averaged variable, the tilde ∼ indicates a Favre-averaged variable, ρ is the air density, U is the velocity vector, P is the air pressure, μ is the dynamic viscosity, u′ is the fluctuating component of the velocity vector, S_M is the momentum source, T is the air temperature, κ is the thermal conductivity, S_T is the source term, and R is the specific gas constant of air.

In the current research, the realizable k-ɛ model that has been proved to be a significant advance over the standard k-ɛ model in dealing with flows featuring strong streamline curvatures, vortices, and rotation²³ is adopted to compute the Reynolds stresses and scalar transport terms, and close the system of mean flow equations (1) to (3). The transport equations for the turbulence kinetic energy k and its dissipation rate ɛ in the realizable k-ɛ model are

div (\bar{ρ} k \tilde{U}) = div [(μ + \frac{μ_{t}}{σ_{k}}) grad k] + 2 μ_{t} S_{ij} \cdot S_{ij} - \bar{ρ} ɛ (1 + \frac{2 k}{γ R \tilde{T}})

(5)

div (\bar{ρ} ɛ \tilde{U}) = div [(μ + \frac{μ_{t}}{σ_{ɛ}}) grad ɛ] + \bar{ρ} C_{1} ɛ \sqrt{2 S_{ij} \cdot S_{ij}} - \bar{ρ} C_{2} \frac{ɛ^{2}}{k + \sqrt{ν ɛ}}

(6)

where

μ_{t} = ρ C_{μ} (k^{2} / ɛ)

is the eddy viscosity in which the variable C_μ depends on the mean strain and rotation rates, the angular velocity of the system rotation, and the turbulence fields,²³ S_ij is the mean rate of deformation of a fluid element, γ is the adiabatic index, ν is the kinematic viscosity, and

C_{1} = max [0.43, (η / η + 5)]

η = (k / ɛ) \sqrt{2 S_{ij} \cdot S_{ij}}

, C₂ = 1.9, σ_k = 1.0, and σ_ɛ = 1.2.

Geometry and mesh generation

The computational domain is the cavity formed by the concentric tube walls that bound the airflow, including the tangential injectors, the channel in the inner tube, the chamber inside the outer tube upstream of the inner tube inlet, and the annular region between the outer and inner tubes, as illustrated in Figure 2. The origin of the Cartesian co-ordinate system adopted is located at the center of the inlet plane of the outer tube, and the z axis is along the tube axis and its positive direction points toward the tube outlets. We investigated the effects of seven structural parameters – the injection angle θ, injector diameter d₁, injector number n, inner tube length l₁, diameter of the outer tube D, inner diameter of the inner tube d₂, and length of the outer tube L, on the tangentially injected swirling flow field inside the concentric tubes with different lengths. For simplicity only one parameter was varied for two or three levels, while the other parameters were kept at fixed values when investigating its effect on the flow pattern. Most of these fixed values are currently common ones used in industry, and the varying levels are the possible alternative values chosen for the design parameters. Therefore, a total of 12 computational cases will be run, as shown in Table 1.

Figure 2.

The schematic diagram of the computational domain.

Table 1.

Tube structure parameters

Case	D[mm]	d₁ [mm]	d₂ [mm]	d₃ [mm]	L [mm]	l₁ [mm]	l₂ [mm)	θ [°]	n
a	4	0.5	1.3	3	20	9	11	35	6
b*	4	0.5	1.3	3	20	9	11	45	6
c*	4	0.5	1.3	3	20	9	11	55	6
d	4	0.3	1.3	3	20	9	11	55	6
e*	4	0.4	1.3	3	20	9	11	55	6
f	4	0.5	1.3	3	20	9	11	55	4
g	4	0.5	1.3	3	20	9	11	55	5
h	4	0.5	1.3	3	20	10	11	55	6
i	5	0.5	1.3	3	20	9	11	55	6
j*	4	0.5	1.1	3	20	9	11	55	6
k*	4	0.5	1.5	3	20	9	11	55	6
l*	4	0.5	1.3	3	22	9	11	55	6

PIV experiments were only carried out for these cases.

The computational domain of the tube system employed in this research, whose upstream portion is of a cylindrical shape, downstream portion is of an annular shape, and the middle portion of the outer tube is intersected by several inclined injectors with smaller diameters, is relatively irregular. Therefore, a multi-block meshing scheme was adopted to mesh the computational domain. In the region where the intersection between the injectors and the middle portion of the outer tube takes place, fine unstructured tetrahedral grids were adopted, while the rest of the computational domain, which is more regular, was discretized using coarser structured hexahedral grids. A non-equilibrium wall function²⁴ was employed for the treatment of the near-wall effect so as to avoid a large number of mesh points to resolve the details of the near-wall region. Grids in adjacent blocks conform at the interface.

Boundary conditions

As seen from the tube structure and computational domain, the flow field contains multiple inlets and outlets with different flow conditions. Since the injector inlets are directly connected to the air reservoir (not shown in Figures 1 and 2), the total pressure at the injector inlets is considered to be equal to that in the air reservoir. The static pressure at the outer tube inlet, outer and inner tube outlets is set to be equal to atmospheric pressure since they are connected to the external atmosphere. A no-slip condition was applied at all of the solid walls. The specifications of the boundary conditions are listed in Table 2.

Table 2.

Boundary conditions

Zone	Boundary condition type	Boundary condition specifications
Injector inlets	Pressure inlet	P₀ = 5 × 10⁵Pa, P_b = 1.01325 × 10⁵Pa, T₀ = 293 K, I = 5%,²⁵ $k = \frac{3}{2} (U_{avg} I)^{2}$ , $ɛ = C_{μ}^{3 / 4} \frac{k^{3 / 2}}{0.07 d_{1}}$
Outer tube inlet	Pressure outlet	P_S = 1.01325 × 10⁵Pa, T₀ = 293 K, I = 5%, $k = \frac{3}{2} (U_{avg} I)^{2}$ , $ɛ = C_{μ}^{3 / 4} \frac{k^{3 / 2}}{0.07 D}$
Outer tube outlet	Pressure outlet	P_S = 1.01325 × 10⁵Pa, T₀ = 293 K, I = 5%, $k = \frac{3}{2} (U_{avg} I)^{2}$ , $ɛ = C_{μ}^{3 / 4} \frac{k^{3 / 2}}{0.07 (D - d_{3})}$
Inner tube outlet	Pressure outlet	P_S = 1.01325 × 10⁵Pa, T₀ = 293 K, I = 5%, $k = \frac{3}{2} (U_{avg} I)^{2}$ , $ɛ = C_{μ}^{3 / 4} \frac{k^{3 / 2}}{0.07 d_{2}}$
Solid walls	No-slip condition	U_s = 0, q_s = 0

P₀, total pressure at the injector inlets; P_b, back pressure for the injectors; P_S, static pressure; T₀, total temperature; I, turbulence intensity; U_avg, mean flow velocity; U_s, air velocity at solid walls; q_s, is heat flux to the walls.

Numerical methods and solution procedure

The numerical simulation was implemented by the ANSYS FLUENT package. The finite volume method was employed to discretize the governing equations on the grid. A second-order upwind differencing scheme was adopted for discretizing the convective terms. In the current research, coupled-implicit formulation, which is more capable of dealing with high-speed compressible flows, was adopted to solve the governing equations of mass conservation, momentum, and energy simultaneously as a group, while governing equations for the turbulence variables k and ɛ were segregated from the coupled set and sequentially solved. The Gauss-Seidel algorithm was employed to solve the discretized algebraic equations in conjunction with the algebraic multigrid (AMG) method.

Grid independence check

The grid was successively refined in order to find the grid-independent solutions. The results for the grid independence check for case c are shown in Figures 3 and 4. The three levels of grids contain 70,457, 166,800, and 466,326 elements, respectively. A refinement ratio of r = 1.4 was chosen for the grid refinement of the three-dimensional model due to the limitation of the performance of the computer used. In spite of this, this refinement ratio was considered to be acceptable.²⁶ The velocity values in Figure 3 have been normalized by dividing them by the theoretical velocity magnitude at the injector exit (see equation (11)). It can be noticed that when the tangential and axial velocities at the same axial position z/R₁ = 4.5 (where R₁ = D/2 is the inner radius of the outer tube) are computed based on the two finer grids they show negligible discrepancies in terms of both their radial distribution characteristics and values. Figure 4 shows the total mass flow rate through the injectors vs. the number of elements. The mass flow rate has been normalized by the theoretical mass flow rate value. It is clearly shown that the two finer grids yield equal results that are independent of the mesh resolution. As a result, the grid level 2 scheme was adopted for the further analysis. The grid generated for case a is shown in Figure 5.

Figure 3.

Radial distribution of the (a) tangential and (b) axial velocities at z/R₁ = 4.5 in case c.

Figure 4.

Normalized total mass flow rate through the injectors vs. the number of elements.

Figure 5.

Grid generated for case a: (a) the whole computational domain; (b) enlarged view.

Experimental

An experiment was conducted to measure the velocity field of the tangentially injected swirling flow inside the concentric tubes of different lengths, to investigate the effects of some structural parameters, and to enable a comparison to be made with the results obtained by the numerical simulation. Various studies have been reported in the literature on measuring the flow field and investigating the characteristics of the swirling flow using experimental methods, including single-point measurements such as Pitot tube,^27–29 hot-wire anemometry (HWA),^30–32 laser Doppler anemometry (LDA),^33–35 and flow visualization.^36–40 As an advanced nonintrusive method that can measure the instantaneous velocity fields, PIV has also been used in the measurement of a swirling flow field.^41–44 However, the experimental measurement of the tangentially injected swirling flow inside the concentric tubes with different lengths had not been reported. The PIV technique was adopted to measure this flow field.

Designing the particle image velocimetry experiment

Since the tube system adopted in this research is an abstracted model of the vortex spinning nozzle used in industry, the size of the tube system is quite small, e.g. the diameter of the outer tube D is only 4 mm, as shown in Figure 2 and Table 1. This indicates that the length scale of the tube system falls between the measurement ranges of PIV and micro-PIV. As a result, it is difficult to directly measure the airflow field inside the actual tube system using the PIV technique. Considering the measurement range of PIV, a magnified tube system with a magnification of 10 is adopted for the measurement, and the geometric similarity is met for the model and the prototype

M_{0} = \frac{L_{m}}{L_{p}} = 10

(7)

where L_m and L_p represent the length scales of the model and prototype, respectively.

Since the Mach number is a key parameter that characterizes compressibility effects in a flow,⁴⁵ Mach numbers were initially matched between the model and prototype, both working with air as the fluid. However, under this condition, some difficulties, or even problems, were met during the PIV measurement. First, the air currents jetted into the outer tube from the injectors mainly flow in the near-wall region, while the air that was sucked into the tube from the outer tube inlet mainly flows in the near-axis region. Therefore, there was a need to seed particles from both the injectors and outer tube inlet. However, a much smaller volume of particles can be sucked into the tube from the outer tube inlet than that jetted into the tube via the injectors, causing a low concentration of particles in the near-axis region and a non-uniform distribution of particles in the tube cross-section. Second, due to the high velocity of the airflow, some particles may enter or move out of the measurement plane between two successive image acquisitions, leading to a bad correlation. Third, due to the higher density ratio of the particles, high speed of the air jets, and the small diameter of the injectors, particles will clog the injectors very quickly and coat the interior surfaces of the tubes due to the centrifugal forces induced by the swirling flow. Therefore, not only will the field of view be blocked but the measurement also has to be paused quite frequently to clean the flow apparatus. Since the structure of the tube system is complex, cleaning is not easy and is time-consuming. Fourth, although an extra container was placed outside the tube outlets to collect the used particles flowing out of the tubes, particles, including those not sucked into the outer tube inlet, would still be dispersed into the ambient air causing contamination to the laboratory. Due to the above reasons, water instead of air was used in the PIV measurement and the Reynolds number instead of Mach number was duplicated between the model and prototype to ensure the dynamic similarity

{Re}_{m} = \frac{ρ_{w} U_{w} L_{m}}{μ_{w}} = {Re}_{p} = \frac{ρ_{a} U_{a} L_{p}}{μ_{a}}

(8)

where the subscripts m and p indicate model and prototype, and w and a indicate water and air. Guo⁴⁴ was also reported to adopt a duplication of the Reynolds number between the model and prototype to ensure dynamic similarity with the model working with water as the fluid for a PIV measurement of the flow field inside an air-jet spinning nozzle with a diameter of only 2 mm. In comparison, the major reason was due to a limited performance of the experimental apparatus.

Experimental apparatus and methods

In order to gain optical access to the internal flow field of the tube system, the model was manufactured using transparent poly(methyl methacrylate) (PMMA) (also known as Plexiglass). Polishing treatment was carried out for the outer and inner surfaces of the model. The outer tube was clamped at each end by two thin metal plates. Each of the metal plates had a stepped hole with two steps, as shown in Figure 6. The large hole had a diameter equal to the outer diameter of the outer tube and the small hole had a diameter slightly larger than the inner diameter of the outer tube. Therefore, the ends of the outer tube can be fixed in the stepped holes of the thin metal plates. A thin screw was used to fix the circumferential position of the outer tube. The inner tube was mounted in a metal support and fixed with a bolt. The two metal plates and the metal support were fixed onto two sliding rails by bolts and their axial positions adjusted according to different test cases. Three cross-sections, z/R₁ = 2.5, z/R₁ = 4.5, and z/R₁ = 6.5 and an axial section x/R₁ = 0 as shown in Figure 6, were chosen for the measurement of each case. In order to avoid obstruction of the injectors in the field of view when measuring the flow field in the axial section, the model was provided with just two injectors arranged with an angular interval of 180°. In order to reduce surface reflections, the outer surfaces of the inner tube, the injectors, and the metal units were all coated with black paint.

Figure 6.

Schematic diagram of the model: (a) front view; (b) side view.

The model was fixed at the bottom of a transparent cuboid-shaped water tank with a size of 100 cm × 40 cm × 30 cm and wall thickness of 10 mm, and completely immersed in water. The axis of the tube system was set parallel to the length direction of the water tank, and the inlet of the outer tube was placed near to one of the end-walls of the tank in order to facilitate the measurement (where the inflow at the outer tube inlet is not affected by the existence of the wall). Water was pumped into the model through the injectors from a water container by a magnetic drive pump whose speed was controlled by a variable-frequency drive. An outlet with a valve was set in the bottom corner on the other end-wall of the tank to connect the tank to the water container through pipes. Thus, the water expelled from the model could flow back into the water container and the water circulated, as shown in Figure 7. The flow rate through the injectors was precisely controlled by an electromagnetic flowmeter and a valve downstream of the pump, while water in the tank freely flowed in and out of the model through the rest of the openings of the model. The water flowing out of the tank was qualitatively controlled by a valve to ensure flow circulation.

Figure 7.

Experimental setup for the PIV measurement.

The flow rate through the injectors of the model as the boundary condition was calculated. According to Table 2, the air velocity U_in,a and density ρ_in,a at the injector exit of the prototype can be calculated by

{Ma}_{in, p} = \sqrt{\frac{2}{γ - 1} [(\frac{P_{0}}{P_{b}})^{(γ - 1) / γ} - 1]}

(9)

T_{in, p} = T_{0} (1 + \frac{γ - 1}{2} {Ma}_{in, p}^{2})^{- 1}

(10)

U_{in, a} = {Ma}_{in, p} \sqrt{γ {RT}_{in, p}}

(11)

ρ_{in, a} = \frac{P_{b}}{{RT}_{in, p}}

(12)

According to equations (8) to (12), the total flow rate at the injector exits of the model can be expressed as

Q_{in, m} = \frac{n π U_{in, w} d_{1, m}^{2}}{4} = \frac{n π d_{1, m}^{2} ρ_{in, a} μ_{w} U_{in, a}}{4 ρ_{w} μ_{a} M_{0}}

(13)

where Ma_in,_p and T_in,p are the Mach number and static temperature at the injector exit of the prototype, respectively, U_in,w is the water velocity at the injector exit of the model, and n and d₁ _, _m are the number and diameter of the injectors of the model, respectively.

The PIV system used for the two-dimensional planar measurement was provided by Dantec Dynamics (Skovlunde, Denmark). The laser was a doubled-pulsed Nd: YAG laser with a maximum repetition frequency of 10 Hz, generating pulses with a wavelength of 532 nm. The laser beam emitted from the laser system was formed into a thin light sheet with a thickness of less than 1 mm by means of an optical lens connected to the laser source by a light-guiding arm. Particle images were captured using a FlowSense EO 4M CCD camera (Dantec Dynamics, Skovlunde, Denmark) with a sensor resolution of 2048 × 2048 and a frame rate of 20.4 fps at full resolution. The CCD camera works with a Nikon AF Micro-Nikkor 60 mm f/2.8D lens (Nikon Corporation, Tokyo, Japan). The positions of both the laser sheet optics and the camera could be precisely adjusted three-dimensionally with a transverse system. During the measurements, the plane of the laser sheet was set coincident with the cross-section for the velocity measurement. The camera axis was placed vertical to the laser sheet (and also the transparent wall of the cuboid water tank). The 80N75 timer box (Dantec Dynamics) was used to synchronize the timing of the laser illumination and the camera recording. Particle images captured by the CCD camera were transferred to the host computer for storage. Data acquisition, analysis, and processing were all controlled and performed using DynamicStudio (Dantec Dynamics, Skovlunde, Denmark) software.⁴⁶ Before the measurements, calibration was done by positioning at the cross-section a calibration target on which there was a regular mesh with a precisely known pitch to obtain the relationship between the corresponding positions in the object domain and the image plane. In the pre-interrogation image processing step, digital masks were applied to eliminate unwanted image signals, such as the images of the tube walls and reflections from them. The adaptive correlation method was used in the interrogation with local validation and high frequency and deforming windows algorithm to remove spurious vectors and achieve sub-pixel accuracy as high as ±0.1 pixel. The interrogation area size was set to 32 × 32. The size of the neighborhood vector area is set to 3 × 3 and the moving average methodology was adopted for the local interpolation of vectors in the local neighborhood validation. The time delay between pulses was determined after careful and repeated adjustment according to the design rules listed in Adrian and Westerweel,⁴⁷ together with the actual flow characteristics at different cross-sections for the measurements.

The water used in the experiment naturally contains a large quantity of small particles. Tests before the experiment showed that the properties, such as the ability to follow the fluid motion and to scatter enough light to create bright images plus concentration of these particles, were quite suitable for the experiment. Therefore, no extra particle seeding was needed for the measurement. Due to the limitation of the experimental conditions, PIV experiments were only carried out for the cases marked * in Table 1.

Results and discussion

Numerical results

Effect of the injector angle

The numerical results for the flow streamline distributions for different injector angles while the other parameters are fixed (cases a, b, and c) are presented in Figure 8. For an injector angle of 35°, the air currents in the section upstream of the injectors flow in a manner almost parallel to the tube axis. After being affected by the air-jets injected from the injectors, they turn into swirling flows and whirl downstream through the annular region between the two tube walls. With the increase of the injector angle, the swirl of the flow upstream of the injectors becomes more intense, while that in the vicinity of the injectors becomes more irregular. The streamlines in the region downstream of the injectors in cases b and c exhibit smaller pitches than that in case a due to larger helical angles. A smaller-pitched streamline in this region indicates an improved efficiency of the airflow twisting the fibers.

Figure 8.

Flow streamlines for the cases with different injector angles. (a) 35°; (b) 45°; (c) 55°.

Figure 9 illustrates the numerical solution for the effect of the injector angle on the radial distributions of the tangential and axial velocities at various axial positions when other tube parameters are at fixed values. Since the radial velocity components are very small compared to the other two components, they are ignored in the following discussion. The velocity values of the simulation are all normalized by the theoretical velocity magnitude U_in,a at the injector exit calculated by equation (11). It can be seen that when the injector angle is 35°, there is no tangential motion for the airflow in the inlet section of the outer tube. The tangential velocity shows small but more fluctuating values than those with larger injector angles in the vicinity of the injector exits. With the increase of the injector angles, the tangential velocity generally increases in the whole region inside the outer tube. However, there is no tangential velocity component for the airflow inside the inner tube for the three cases. For the axial velocity, the air current flows into the outer tube with neither varying values along the radial direction nor reverse flow in the near-wall region when the injector angle is 35°. The highest value in the annular region is noticeable, while the reserve flow inside the inner tube is the weakest. These are beneficial for drawing fibers into the twisting system at the start of the spinning process. Reverse flow was observed near the wall of the inlet region of the outer tube when the injector angle is 45°, and its intensity increases with a further increase of the injector angle. These can also be seen in Figure 8. When the injector angle was 55°, the intensity of the reverse flow inside the inner tube also reached its highest value of the three cases. This mainly resulted from the lowest air pressure in the core region of the swirl generated at the largest injector angle. Therefore, the key to optimizing the injector angle is to balance between the efficiency of twisting fibers and the facilitation of drawing fibers into the twisting system. In another perspective, the compromise between these two factors will lead to the capping of the vortex yarn tenacity.

Figure 9.

Effect of injector angle on the tangential and axial velocities of the tangentially injected swirling flow. (a) Normalized tangential velocity; (b) normalized axial velocity.

Effect of the injector diameter

Figure 10 presents the numerical results for the effect of the injector diameter on the radial distribution of the tangential and axial velocities at various axial positions when the other parameters are fixed (cases c, d, and e). It can be clearly observed that the tangential velocity value increases with the enlargement of the injector diameter. In particular, the increasing trend was much greater when the injector diameter is increased from 0.3 mm to 0.4 mm than when it was increased from 0.4 mm to 0.5 mm. Only marginal increases of the tangential velocity can be observed when the injector diameter is increased from 0.4 mm to 0.5 mm. Close to the injector exits, the radial position for the maximum tangential velocity moves toward the tube axis, indicating an enlargement of the free-vortex region. In the annular region, the axial velocity also increased with the increase of the injector diameter. Noticeable reverse flow exists in the inlet region of the outer tube and inside the inner tube when the injector diameter is 0.5 mm. Taking together the tangential and axial flow characteristics obtained from the simulation, an injector diameter of 0.4 mm is more reasonable under the current parameter settings. This is in agreement with the results obtained by Shang et al.⁴⁸ from conducting a spinning experiment and by Pei and Yu⁴⁹ modeling the fiber dynamics in the vortex spinning nozzle. It has been reported from industry that injectors with diameters of 0.5 mm and 0.6 mm were adopted by the Murata No. 861 MVS spinner (Murata Machinery Ltd.).⁵⁰ Our results could be a good supplement to the methods and results for the optimization of the injector diameter of the vortex spinning nozzle.

Figure 10.

Effect of injector diameter on the tangential and axial velocities of the tangentially injected swirling flow. (a) Normalized tangential velocity; (b) normalized axial velocity.

Effect of the number of injectors

Figure 11 shows the numerical results for the effect of the number of injectors on the radial distribution of the tangential and axial velocities at various axial positions when the other parameters are fixed (cases c, f, and g). When the number of injectors was increased from four to six, the tangential velocity values increased slightly. As for the axial velocity, the increase in number of injectors also led to a slight increase of this velocity component of the swirling flow in the annular region between the two tubes and the axial reverse flow inside the inner tube. Influenced by this intense reverse flow, an obvious fluctuation of the axial velocity, which is not good for the orderly motion of fibers, can be observed in the core region near the injector exits of the outer tube. Generally, one can see that the influence of the number of injectors is marginal for both tangential and axial velocity values. Therefore, taken together with the flow characteristics and the energy consumption, a higher number of injectors is not obviously beneficial for an improved performance of the airflow. It has been reported that four or five injectors have been used industrially by the Murata No. 861 MVS spinner, with four injectors usually employed for the economical-type nozzle for saving air consumption.⁵⁰ The analysis above in regard to the airflow pattern could be a good explanation for the design adopted by the machinery manufacturer.

Figure 11.

Effect of the injector number on the tangential and axial velocities of the tangentially injected swirling flow. (a) Normalized tangential velocity; (b) normalized axial velocity.

Effect of the inner tube length

In our model, the inner tube represents the leading tip of the hollow spindle inserted into the nozzle block. The numerical results for the effect of the inner tube length on the radial distribution of the tangential and axial velocities at various axial positions when the other parameters were fixed are shown in Figure 12 (cases c and h). It can be seen that although the length of the inner tube varies, both the velocities in the vicinity of the injector exits and in the annular region between the tubes are hardly influenced. The tangential velocity of the case with an inner tube length of 9 mm shows higher values than the case of 10 mm in the near-wall region of the inlet portion of the outer tube. As for the axial velocity, higher values can be observed in the core region of the inlet portion of the outer tube with a shorter inner tube length. In addition, a shorter inner tube length also led to higher axial velocities of the reverse flow inside the inner tube. These may result from a lower air pressure in the core region of the outer tube caused by a stronger swirl under a weaker resistance offered by a shorter inner tube length. However, the differences between the axial velocity components in these regions are not so obvious. According to the claims made by the machinery manufacturer⁵⁰ and some researchers,⁵¹ a shorter length of the hollow spindle tip being inserted into the nozzle results in more wrapping fibers in the yarn. In view of the airflow characteristics, it seems that the flow pattern is less affected by this parameter. Taking together these two factors – the number of wrapping fibers and the flow pattern – a higher yarn tenacity could usually be obtained with a shorter length of hollow spindle tip being inserted into the nozzle, which is in agreement with the conclusion drawn by Pei and Yu,⁴⁹ whose analysis was based on the fiber dynamics in the nozzle.

Figure 12.

Effect of the inner tube length on the tangential and axial velocities of the tangentially injected swirling flow. (a) Normalized tangential velocity; (b) normalized axial velocity.

Effect of the outer tube diameter

The outer tube corresponds to the nozzle block, while the internal cavity of the outer tube corresponds to the air chamber. Figure 13 presents the numerical solution for the effect of outer tube diameter on the radial distribution of the tangential and axial velocities at various axial positions when the other parameters were fixed (cases c and i). An air chamber diameter of 4 mm is usually adopted in industry by machinery manufacturers, while a diameter of 5 mm is an alternative for investigation in the current research. Upstream of the injectors, the tangential velocity of the airflow decreases greatly with the increase of the outer tube diameter. Downstream of the injectors, there are no obvious changes to the value of the tangential velocity while its distribution is radially expanded. This in turn leads to a large decrease of the tangential velocities in the core region of the air chamber, which will deteriorate the twisting efficiency in this area. Therefore, the design of the air chamber of the vortex spinning nozzle needs to take the coarseness of the fiber strand (or the yarn count) into consideration. In view of the fiber twisting, an air chamber with a larger diameter is not suitable for a vortex yarn of fine count. The effect of the outer tube diameter on the axial velocity is more complicated. The reverse flow in the near-wall region of the inlet portion of the outer tube almost disappears when D increases from 4 mm to 5 mm. In the core region, the axial velocity exhibits a lower value with D = 5 mm than with D = 4 mm at z = 1 mm, while it exhibits a larger value at z = 5 mm. At D = 5 mm, there was an obvious decrease of the axial velocity value for the reverse flow inside the inner tube when compared to D = 4 mm. As a result, the reverse component for the axial flow in the core region at z = 9 mm disappeared. With an increase of the cross-sectional area in the annular region, the axial velocity decreased due to conservation of the mass flow rate.

Figure 13.

Effect of the outer tube diameter on the tangential and axial velocities of the tangentially injected swirling flow. (a) Normalized tangential velocity; (b) normalized axial velocity.

Effect of the inner diameter of the inner tube

The cavity formed inside the inner tube corresponds to the yarn delivery passage of the hollow spindle. Figure 14 shows the numerical results for the effect of the inner diameter of the inner tube on the radial distribution of the tangential and axial velocities at various axial positions when the other parameters are fixed (cases c, j, and k). Besides the slight increase of the axial velocity inside the inner tube, an increase of the inner diameter of the inner tube also had an effect on the flow in the outer tube. The axial velocity upstream of the injectors shows a decreasing trend when d₁ increases. This may be due to an increased resistance generated by the reverse flow inside the inner tube, which is suggested by the variation of the axial velocity values at z = 9 mm. The axial velocities in the annular region between tubes were hardly influenced by the inner diameter of the inner tubes. For the tangential velocities, similar values in the upstream region of the injectors at d₁ = 1.1 mm and d₁ = 1.3 mm can be observed. In the downstream region, the tangential velocity values are quite similar for the three cases. In the vicinity of the injector exits, the variation of the inner diameter of the inner tube had no influence on the tangential velocity. Therefore, it can be seen that the twisting of fibers will be less affected by the inner diameter of the hollow spindle in the perspective of the airflow characteristics.

Figure 14.

Effect of the inner diameter of the inner tube on the tangential and axial velocities of the tangentially injected swirling flow. (a) Normalized tangential velocity; (b) normalized axial velocity.

Effect of the length of the outer tube

Figure 15 presents the numerical results for the effect of the outer tube length on the radial distribution of the tangential and axial velocities at various axial positions when the other parameters were fixed (cases c and l). In these two cases, the injector exits were both located at z = 9 mm. The origin of the z axis lay on the inlet plane of the tube with L = 20 mm. The inlet plane of the tube with L = 22 mm lay at z = −2 mm. Therefore, the variation of the outer tube length denotes the different lengths of the inlet section of the nozzle block. From this figure, we can see that the variation of the outer tube length had no influence on the velocities either in the annular region between the two tubes or inside the inner tube. At the cross-section of z = 1 mm, with the increase of outer tube length, the tangential velocity in the near-axis zone increased while the axial velocity decreased with an enlargement of the reverse flow zone. In the vicinity of the injector exits, i.e. z = 9 mm, the increase of the outer tube length caused a shift of the radial distribution of the velocities, while their values show little change. The simulation results indicate that, similar to the inner diameter of the inner tube, the length of the outer tube is less influential upon the twisting of fibers during the spinning process.

Figure 15.

Effect of the outer tube length on the tangential and axial velocities of the tangentially injected swirling flow. (a) Normalized tangential velocity; (b) normalized axial velocity.

Qualitative comparison of numerical and experimental results

The PIV experimental results for the flow characteristics of the tangentially injected swirling flow inside the concentric tubes with different lengths are presented in Figure 16. The measured velocity values have all been normalized by U_in,w. PIV measurements in the cross-sections of z/R₁ = 2.5, z/R₁ = 4.5, and z/R₁ = 6.5 simultaneously give the tangential and radial velocities. At the cross-section of z/R₁ = 4.5, where the injector exits are located, highly swirling flow was generated in the region far away from the tube axis (near-wall region). A high-velocity zone existed downstream of the exit of each injector. The fluid velocity gradually decreased when the fluid flows downstream away from the exit and the high-velocity zone expanded radially toward the tube axis. Each high-velocity zone circumferentially occupied about 1/4 of the tube. This can be the explanation for our numerical result that four injectors are enough for generating an effective swirling flow. The readers should bear in mind that the reason for adopting only two injectors in the experiment is for facilitating the measurement of the velocity on the axial section. In the radial direction, the high-velocity zone occupied about 1/2 of the tube, which is also in accordance with the numerical results. In the near-axis region of the tube, the fluid velocity was quite small and the swirl of the flow is not obvious. Several small and irregularly distributed eddies can be observed in this region. These irregularly distributed small eddies may be generated as a result of the collision between the fluid flowing into the outer tube from the outer tube inlet and the reverse flow from the inner tube. At the cross-section of z/R₁ = 6.5, which lies in the region between the inner and outer tubes, due to the shading of light by the inner tube, the fluid velocity in the region below the inner tube was not obtained in the experiment. In this annular section, regular swirling flow around the inner tube could be observed. The maximum value of the velocity was lower than that at the cross-section of z/R₁ = 4.5, indicating that the swirling flow was decaying. Nevertheless, the fluid velocities at the cross-section of z/R₁ = 6.5 were more evenly distributed. This is due to the mixing and entrainment of the tangential injection downstream of the injectors. The swirl of the flow was only observed in a small near-axis zone with an irregular shape in the inlet region of the outer tube, e.g. at the cross-section of z/R₁ = 2.5; and the fluid velocities at this cross-section are quite small, indicating the flow in the inlet region of the outer tube was not greatly influenced by the downstream swirl. The measured velocity at the axial section (x/R₁ = 0) of the outer tube is also shown in Figure 16. Unfortunately, due to the reflection of light by the tube walls, neither the velocities in the near-wall region of the outer tube nor the annular region between the outer and inner tubes could be obtained in the PIV experiment. Therefore, only the velocities in the near-axis region in the outer tube are shown. Axial and radial velocity components of the flow could be obtained simultaneously in this axial section. The measured results show that the radial velocity components of the flow in the near-axis zone in the inlet region were not obvious, and the fluid flowed downstream nearly parallel to the axis of the tube. In the vicinity of the upstream end of the inner tube, the radial velocity components of the fluid gradually increased, showing the fluid in the near-axis region flowing outward into the annular region between the inner and outer tubes. With the fluid flowing downstream approaching to the inner tube inlet, the axial velocity gradually decreased. Besides, in the region near the inner tube inlet, the value of the axial velocity gradually decreased with the decrease of the radial distance from the tube axis. Both of these characteristics are the direct results caused by the reverse flow inside the inner tube.

Figure 16.

Experimental results of the velocity vectors and streamlines with contour plots of velocity magnitude for different cases: (a) z/R₁ = 2.5 (Case b); (b) z/R₁ = 4.5 (Case e); (c) z/R₁ = 6.5 (Case e); (d) z/R₁ = 0 (Case c).

From Figure 16 it can be seen that the fluid velocities measured by the PIV experiment, which adopted the duplication of the Reynolds number between the model and prototype using water as the working medium, are lower than the airflow velocities in the prototype obtained by the numerical simulation. This can be further demonstrated in Figure 17, showing the qualitative comparison of the tangential and axial velocities at different cross-sections between the numerical simulation and the experiment. The measured tangential velocities were extracted by interpolation along the radial direction where the maximum value exists using TECPLOT (Tecplot, Inc., Bellevue, WA, USA) software. Although, as expected, discrepancies were observed between the numerical and experimental results, the distribution characteristics of the velocities and their trends influenced by the structural parameters coincide with each other. In Figure 17(a), the numerical and experimental results for the tangential velocities at the cross-section of z/R₁ = 4.5 both exhibit small values with flat distributions extending to a radial range on either side of the tube axis. Then, with a further increase of the radial distance, a sharp increase is observed on both sides, showing a typical characteristic of swirling flow. In addition, both the numerical and experimental results for the case with an injector angle of 45° (case b) exhibit the lowest tangential velocity values compared to the other cases. As a result of the compressibility of the high-speed air, expansion waves were generated when the air currents flowed into the outer tube from the injectors, leading to a great increase of the airflow velocity values at the cross-section of z/R₁ = 4.5 in the prototype.^15,52 Due to the higher viscosity, the water flow decelerated after it was discharged from the injectors and whirled downstream. This must be the main reason for the discrepancy between the numerical and experimental results. Similar comparison results were observed for the tangential velocities at the cross-section of z/R₁ = 6.5, which exhibited a rapid increase from zero, with the radial distance increasing from the inner tube wall, and then a flat distribution with relatively stable values, followed by a rapid decrease to zero with the radial distance further increasing towards the outer tube wall, as shown in Figure 17(b). As for the axial velocity, although no negative values of the experimental results were observed in the near-axis region at the cross-section of z/R₁ = 4.5, its distribution characteristics are in accordance with the numerical results with increasing values from the minimum in the near-axis zone to the maximum near the wall, as can be observed in Figure 17(c). The absence of the negative axial velocity in the near-axis region in the experimental results may be due to the weaker reverse flow generated inside the inner tube, which is longer than that employed in the numerical simulation in order to facilitate the mounting of the inner tube. The radial distribution of the axial velocity at the cross-section of z/R₁ = 2.5 is shown in Figure 17 (d). It can be observed that both numerical and experimental results show higher values in the near-axis zone than those at the radial positions approaching to the outer tube wall. The reverse flow in the near-wall region obtained by the numerical simulation was not duplicated by the experiment due to the limitation of the current PIV measurement. Qualitatively matched results for the effects of structural parameters on the axial velocities at both cross-sections obtained by the numerical simulation and the PIV experiment can be observed.

Figure 17.

Qualitative comparison of the velocities at different cross-sections between the numerical simulation and experiment. Tangential velocity: (a) z/R₁ = 4.5; (b) z/R₁ = 6.5; axial velocity: (c) z/R₁ = 4.5; (d) z/R₁ = 2.5.

Conclusions

The tangentially injected swirling flow in concentric tubes with different lengths that serve as a model of the vortex spinning nozzle was numerically simulated. The realizable k-ɛ model was adopted to model the turbulence. Based on the simulation, the effects of some structural parameters on the swirling flow characteristics in the tubes were analyzed. In the investigation, only one parameter was changed at a time when the other ones were fixed. The PIV technique was adopted to measure the velocity field in the concentric tubes based on a duplication of Reynolds number between the tube model and prototype. Numerical and experimental results were qualitatively compared. The conclusions of this study can be drawn as follows.

Numerical simulation reveals that the increase of the injector angle strengthens the swirl of the flow both upstream and downstream of the injectors and disturbs that in the vicinity of the injectors, which improves the efficiency of twisting fibers while going against the drawing of fibers into the twisting system, leading to a capping of the vortex yarn tenacity. Tangential velocity increases when the injector diameter increases from 0.3 mm to 0.5 mm with only a marginal increase observed and noticeable reverse flow generated when it increases from 0.4 mm to 0.5 mm. The influence of the injector number is marginal for both tangential and axial velocities, and four injectors are enough for generating an evenly distributed swirling flow with the desired intensity and lowered energy consumption in the concentric tubes. An increased outer tube diameter leads to a considerable decrease of the tangential velocity in the core region due to the radial expansion of the airflow and deteriorates the twisting efficiency in this area. A match between the diameter of the air chamber of the vortex spinning nozzle and the coarseness of the fiber strand (or the yarn count) needs to be considered. The other parameters like the inner tube length, inner diameter of the inner tube, and outer tube length are less influential upon the airflow patterns.

The PIV experiment, which adopted a duplication of the Reynolds number for the dynamic similarity between a test model and the prototype, was found to be able to qualitatively replicate many of the characteristics of the tangentially injected swirling flow in the concentric tubes with different lengths. Highly swirling flow with high velocities exists in the near-wall region in the vicinity of the injector exits. Each high-velocity zone occupies about 1/4 of the tube circumferentially and 1/2 in the radial direction. In the region near the inner tube inlet, the axial velocity gradually decreases with the decrease of the radial distance from the tube axis. The distribution characteristics of the velocities and their trends influenced by the structural parameters measured by the PIV technique coincide with the numerical simulation, despite the discrepancy between the velocity values provided by the two methods due to the compressibility effects of the airflow in the prototype.

The future extension of this work is to investigate the interactive effects of these parameters to carry on more comprehensive optimization and to develop experimental methods that can quantitatively reproduce the velocity field of the high-speed swirling airflow.

Footnotes

Acknowledgements

The authors are grateful to Professor Zhenhua Lu, Mr Hao Jin and Mr Liangliang Sun of the Key Laboratory for Power Machinery and Engineering of Ministry of Education, and Mr Qiankun Ma of Shanghai Jiaotong University for their support and assistance with the PIV experiment.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 11302048), the Shanghai Natural Science Fund (grant number 12ZR1440300), Specialized Research Fund for the Doctoral Program of Higher Education (grant number 20130075120002), and Textile Vision Science & Education Fund and China Textile Industry Federation Basic Application Research Fund and the Fundamental Research Fund for the Central Universities of China (grant number 2232013D3 - 06).

Nomenclature

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Case	D[mm]	d₁ [mm]	d₂ [mm]	d₃ [mm]	L [mm]	l₁ [mm]	l₂ [mm)	θ [°]	n
a	4	0.5	1.3	3	20	9	11	35	6
b*	4	0.5	1.3	3	20	9	11	45	6
c*	4	0.5	1.3	3	20	9	11	55	6
d	4	0.3	1.3	3	20	9	11	55	6
e*	4	0.4	1.3	3	20	9	11	55	6
f	4	0.5	1.3	3	20	9	11	55	4
g	4	0.5	1.3	3	20	9	11	55	5
h	4	0.5	1.3	3	20	10	11	55	6
i	5	0.5	1.3	3	20	9	11	55	6
j*	4	0.5	1.1	3	20	9	11	55	6
k*	4	0.5	1.5	3	20	9	11	55	6
l*	4	0.5	1.3	3	22	9	11	55	6

Case	D[mm]	d₁ [mm]	d₂ [mm]	d₃ [mm]	L [mm]	l₁ [mm]	l₂ [mm)	θ [°]	n
a	4	0.5	1.3	3	20	9	11	35	6
b*	4	0.5	1.3	3	20	9	11	45	6
c*	4	0.5	1.3	3	20	9	11	55	6
d	4	0.3	1.3	3	20	9	11	55	6
e*	4	0.4	1.3	3	20	9	11	55	6
f	4	0.5	1.3	3	20	9	11	55	4
g	4	0.5	1.3	3	20	9	11	55	5
h	4	0.5	1.3	3	20	10	11	55	6
i	5	0.5	1.3	3	20	9	11	55	6
j*	4	0.5	1.1	3	20	9	11	55	6
k*	4	0.5	1.5	3	20	9	11	55	6
l*	4	0.5	1.3	3	22	9	11	55	6

Case	D[mm]	d₁ [mm]	d₂ [mm]	d₃ [mm]	L [mm]	l₁ [mm]	l₂ [mm)	θ [°]	n
a	4	0.5	1.3	3	20	9	11	35	6
b*	4	0.5	1.3	3	20	9	11	45	6
c*	4	0.5	1.3	3	20	9	11	55	6
d	4	0.3	1.3	3	20	9	11	55	6
e*	4	0.4	1.3	3	20	9	11	55	6
f	4	0.5	1.3	3	20	9	11	55	4
g	4	0.5	1.3	3	20	9	11	55	5
h	4	0.5	1.3	3	20	10	11	55	6
i	5	0.5	1.3	3	20	9	11	55	6
j*	4	0.5	1.1	3	20	9	11	55	6
k*	4	0.5	1.5	3	20	9	11	55	6
l*	4	0.5	1.3	3	22	9	11	55	6