Oh, T H
ABSTRACT
This paper reports a numerical analysis of transonic flows in the axisymmetric backward-facing step main nozzle of an air-jet loom. To obtain basic design data for the optimum main nozzle shape of an air-jet loom and to predict transonic/supersonic internal flows, a characteristic-based, upwind flux difference-splitting, compressible Navier-- Stokes method is used. Wall static pressure and flow velocity distributions in the nozzle are analyzed by changing air tank pressures and acceleration tube lengths. The flow inside the nozzle experiences double choking, first at the needle tip and then at the acceleration tube exit at air tank pressures near 4 kg^sub f^/cm^sup 2^. The air tank pressure that leads to critical conditions depends on acceleration tube length, i.e., higher air tank pressures for longer acceleration tubes. The air pressure required to bring the acceleration tube exit to sonic conditions is nearly constant regardless of acceleration tube length. The round needle tip shape could lead to less total pressure loss when compared with step shape.
An air-jet loom inserts the weft into the warp by using high pressure air-jet thrust force and skin friction force along the yarn. The air-jet loom is popular in texturing industries because of its high productivity, convenient controllability, and wide variety of textured fabrics: silk, cotton, wool, and spun textures. It causes no air pollution since it uses air as the yarn carrying medium. The air-jet loom is also capable of texturing spun or cellulose filament fabrics, which cannot be woven by water-jet looms. However, since the density of air is too low (about 1/1000 of water) compared with that of water, the compressed air jet diffuses rapidly into the atmosphere after discharging from the acceleration tube. Also, the viscosity of air is about 1/50 that of water, so air consumption becomes critical in an air-jet loom.
Due to this large consumption of air and compressor electricity, increased manufacturing costs are one of the loom's disadvantages. Main nozzle shape, exit shape of subnozzles, response time of the solenoid valve, body shape, subnozzle locations, and control methods have been studied intensively to reduce air consumption in air-jet looms.
Fundamental research in main nozzle flows is especially necessary to design optimum shapes of main nozzles, which push the weft to fly through the warp. The air jet from the main nozzle is easily diffused, and it is hard to control the flow direction and velocity, thus increasing air and compressor electricity consumption. Reduced air-jet diffusion and effective control of flow direction and velocity are therefore very important in air-jet loom design.
Due to the recent development of high speed air-jet looms, studies of transonic/supersonic flows in main nozzles have become important for performance enhancement of air jets and optimum nozzle design. It is very difficult to measure the flow field near the nozzle throat region experimentally due to its small cross-sectional area; however, the most important flow phenomena, such as shock waves and flow separations, occur frequently inside this nozzle throat area. Therefore, a computational analysis of the flow field inside the air-jet main nozzle is necessary.
Air flow inside the main nozzle shows subsonic, transonic, and supersonic flow characteristics. There are some difficulties in computational analysis due to the complexity of the flow nature, i.e., turbulence and shock wave/boundary layer interaction. The governing Navier-- Stokes equations are mixed elliptic/hyperbolic partial differential equations [4].
In previous studies of the air-jet loom, Duxbury and Lord [2] derived air-jet velocity distribution exposed to free atmosphere from the main nozzle. Lyubovitskii [7] measured supersonic pulsed jet flow from the nozzle exit. Uno and Ishida [11] experimentally measured air-jet velocity and weft flying distance using various acceleration tube prototypes. Mohamed and Salama [8] studied the effects of the diameter and length of acceleration tubes on flow velocity. Researchers investigated the converging-diverging nozzle (Kim [4], Mohamed and Salama [8]) flows both experimentally and numerically, but there has not been much air-jet nozzle flow analysis. Ishida and Okajima [3] measured pressure inside the main nozzle experimentally and obtained qualitatively similar results in the acceleration tube by using a one-- dimensional Fanno flow assumption. In their case, they did not obtain detailed information. If we use the full Navier-Stokes equation to analyze the internal transonic/ supersonic flow, we may be able to explain the complex flow phenomena of shock waves and flow separations in detail.
Since the flow inside the main nozzle is typically an axisymmetric, compressible, viscous, and transonic/supersonic flow, a computational fluid dynamics (CFD) approach using the full Navier-Stokes equations is necessary. This approach is especially important in the early design stage, because cFD can be used to analyze the main nozzle flow effectively with minimum costs and efforts before making experimental scale models.
We use the CSCM upwind compressible Navier-Stokes method of Lombard et al. [6] in this study. This method has the merits of an upwind scheme, ease in applying characteristic boundary conditions, and a fast flow solver using a diagonally dominant ADI. Kwon et al. [5] and Song et al. [10] analyzed the transonic/supersonic compressor cascade flow and the performance of transonic centrifugal compressor diffusers, respectively, using this same method.
In this study, we analyze transonic/supersonic viscous flows inside the nozzle using the inlet nozzle flow conditions from the experiments. We also analyze the effect of air tank pressure, acceleration tube length, and circular arc radius change in the backward-facing step on the flow field. Finally, we investigate flow physics, including choking phenomena at the nozzle throat and at the exit of the acceleration tube, and optimum nozzle shapes for proper weft insertion.
Numerical Analysis
GRID SYSTEM AND BOUNDARY CONDITIONS
A 180 X 70 H-type grid system was generated by an elliptic PDE grid generator. Grids were packed near the wall by a stretching function (Figure 2), and then in order to observe free jet flow outside the tube exit, a 20 X 120 grid system was attached to the internal grid system.
As wall boundary conditions, a no-slip boundary condition was prescribed at the walls, ie., u = 0 and v = 0. The inlet and exit boundary conditions depended on flow conditions, i.e., whether flow was subsonic or supersonic.
For supersonic inflow conditions, all the upstream conditions, i.e., flow direction, velocity, total enthalpy, and entropy, were specified. At the supersonic exit, all flow variables were extrapolated from the inside numerically. Meanwhile, inflow direction, entropy, and total enthalpy for subsonic inlet flow and exit static pressure for subsonic outflow were specified. The exit static pressure was specified as atmospheric pressure (101.3kPa) in this study. An adiabatic wall temperature boundary condition was used at the wall.
Results and Discussion
Flow passage in the main nozzle of the air-jet loom is divided into three regions. The first region includes an air tank, a two-way solenoid valve, an air-jet main nozzle inlet, an inclined flow passage, and a minimum cross-- sectional area at the leading edge of the needle. The flow is accelerating in this convergent nozzle region.
Sudden expansion in the nozzle cross-sectional area causes strong flow expansion, and the accelerated flow pulls the weft into the acceleration tube in this second flow region. A massive flow separation right behind the backward-facing step can be observed.
In the acceleration tube, the cross-sectional area is constant, and due to wall friction, the boundary layer develops continuously and eventually the flow with sufficient air pressure is accelerated to Mach I at the exit of the acceleration tube in the third flow region, the "Fanno flow" region. The air jet along with the weft releases freely from the exit of the acceleration tube to the atmosphere and the weft flies into the warp in the air-jet loom.
AIR-JET FLOWS AND GRID CONVERGENCE TESTS
We have performed grid convergence tests by varying the number of grids extensively. As a typical test condition, the air tank stagnation pressure and temperature are 2 kg^sub f^/cm^sup 2^ and 296K, respectively. The inlet Mach number is 0.43, and inlet pressure and the speed of sound are computed from a perfect gas equation of state and isentropic flow conditions. The Reynolds number, based on nozzle inlet velocity and the radius of the acceleration tube (R = 2.0mm), is 3.57 X 10^sup 4^. Thus, the flow inside the nozzle is fully turbulent. Detailed flow conditions are given in Table I.
Figure 1 shows a schematic diagram of the main nozzle system of the air-jet loom used in our study. Major specifications are the inner diameter of the needle (d^sub i^ = 2.8 mm), acceleration tube length (L = 270 mm), and tube diameter (D = 4.0 mm).
Pulsed air from the solenoid valve is accelerated through a narrow stabilizer and flows into the main nozzle. In this analysis, we assume the air flows steadily into the nozzle instead of pulsing. Computation begins 13 mm upstream of the nozzle throat where the flow is parallel to the axial direction. The main nozzle has a hollow needle (or yarn tube) in the middle for weft insertion, and the co-axial surrounding flow passage is quite different from traditional nozzles.
Figure 2 shows the computational grid system. Since we assume an axisymmetric nozzle, we have constructed the upper half grid system only. We treat the hollow needle as a solid one to simplify the complex flow field in a way similar to Mohamed and Salama [8]. Thus, we assume the main nozzle system is an axisymmetric backward-facing step with the acceleration tube apart. All the lengths are nondimensionalized by the radius of the acceleration tube (R = 2.0mm).
Figure 3 shows the grid convergence test results: pressure distribution along the center line at a reservoir stagnation air pressure of 2 kg^sub f^/cm^sup 2^. We use 140 X 70, 150 X 70, 160 x 70, 170 X 70, 180 x 70, and 190 X 70 grids to study the grid effect. The center-line pressure distributions from various grids are almost identical. As shown in Figure 4, separation lengths based on the reattachment point change only 2% when the grid numbers are doubled. The number of grid points under consideration has no significant effect on the computational resuits, therefore, we used the 180 X 70 grid system for all computations.
EFFECT OF AIR TANK PRESSURE
Figure 5(a-e) shows pressure contours in the sudden expansion zone near the backward-facing step using Ishida and Okajima's test conditions [3] at air tank pressures of 2-6 kg^sub f^/cm^sup 2^, respectively. The flow can be accelerated from subsonic to supersonic at the throat of the convergent-- divergent nozzle. The portion behind the nozzle throat is a sudden expansion zone where complex flow patterns such as turbulent air jets and recirculating flows exist. Pressure decreases rapidly right after the throat, and subsequently increases before the acceleration tube inlet. Thus, pressure contours in this region are also extremely complex. Due to the sudden expansion behind the nozzle throat, there are low supersonic flow regions at 2 and 3 kg^sub f^/cm^sup 2^ air tank pressures; however, flow choking (the maximum mass flow rate possible through the nozzle throat, which could occur at sonic speed) does not occur at the throat. At air tank pressures over 4 kg^sub f^/cm^sup 2^ there is sonic flow (M = 1) at the throat, and it shows similar flow patterns in pressure contours near the backward-facing step zone.
Figure 6 shows Mach contours along the center line at tank pressures from 2 to 6 kg^sub f^/cm^sup 2^. Air jet velocities at the acceleration tube exit are subsonic at 2 and 3 kg^sub f^/cm^sup 2^ and sonic, i.e., choked flow, over 4 kg^sub f^/cm^sup 2^. Due to the choked flow at the acceleration tube exit, the flow patterns, i.e., Mach number distributions, do not change as the tank pressures increase over 4 kg^sub f^/cm^sup 2^.
Static wall pressure distributions nondimensionalized by inlet static pressure at various air tank pressures are shown in Figure 7. Nondimensional wall pressure (p/ p^sub ref^) decreases gradually at 2 kg^sub f^/cm^sup 2^ and reaches nearly choked flow conditions at 3 kg^sub f^/cm^sup 2^ and choked flow conditions at 4 kg^sub f^/cm^sup 2^ or higher, thus showing almost identical values from inlet to exit. The acceleration tube exit pressure reaches atmospheric pressure at 2 and 3 kg^sub f^/cm^sup 2^; however, at 4 kg^sub f^/cm^sup 2^ or higher air tank pressures, exit velocities become sonic and exit pressures are higher than atmospheric pressure.
If we assume the optimum air tank pressure as one that achieves sonic velocity at the acceleration tube exit with minimum air tank pressure, then the optimum pressure in the main nozzle system is close to 4 kg^sub f^/cm^sup 2^. Pressure higher than 4 kg^sub f^/cm^sup 2^ is not necessary, and lower pressure than this does not accelerate the flow effectively.
Figure 8 shows Mach number distribution along the center line in the full computational domain. The center line Mach number decreases rapidly as air-jet flows out of tube exit into the surrounding air.
Figure 9 shows a comparison of a few nondimensional velocity profiles along the path of the free jet at 3 and 5 kg^sub f^/cm^sup 2^ to a Gaussian distribution function. Y and V are nondimensionalized by the jet radius, where the magnitude of velocity is a quarter of the maximum velocity and the maximum velocity at each position, respectively. Velocity profiles along the direction normal to the nozzle axis are self-similar over X/D = 40. These normalized velocity profiles are similar to a Gaussian function, so we can assume that our numerical results are valid when compared with the analytic method.
EFFECT OF ACCELERATION TUBE LENGTH
We have studied the effect of acceleration tube length on the flow field by varying tube lengths (L = 180, 240, and 270 mm) at various air tank pressures. Figure 10 shows the Mach number distribution along the center line at a tank pressure of 3 kg^sub f^/cm^sup 2^. The change in tube length does not cause any significant flow pattern changes near the nozzle throat. Mach numbers increase slowly along the tube and at the tube exit rise slightly as tube length increases, but the changes in exit Mach numbers are very small. Figure 11 shows pressure distributions along the center line with various lengths (L = 180, 240, 270 mm) at an air tank pressure of 3 kg^sub f^/cm^sup 2^. Regardless of tube lengths, the exit pressures are similar to each other. When the tube is long, the mechanical energy loss inside the tube becomes large compared with a short tube. Since dimensionless exit pressures are almost the same, the dimensionless tube inlet pressure (10
The thrust force pulling the yam is generally proportional to yarn-flow contact distance and flow velocity. The longer the acceleration tube, the better flow stability and thrust force. In this regard, the optimum length should be properly considered.
EFFECT OF NOzzLE SHAPE
We used a square backward-facing step in the computations above. In this section, we changed the square backward-facing step to circular arc shapes of different radii to see how flow changes in the nozzle system. The radii (r/R) of the circular arcs considered in this study are 0.0625, 0.125, and 0.1875.
Figure 12 shows pressure distributions along the center line at 5 kg^sub f^/cm^sup 2^. The circular arc shape nozzles show higher static pressures compared with the square shaped nozzle. The largest radius of a circular arc, r/R = 0.1875, has the largest static pressure. Therefore, we know that shape changes inside the nozzle using a circular arc sustain higher static pressures inside the acceleration tube. By rounding the square edge of the backward-facing step, we can reduce total pressure loss inside the nozzle. Thus, we can obtain sonic conditions at the tube exit at slightly lower than 4 kg^sub f^/cm^sup 2^, which reduces air consumption by a few percent. The separation zone lengths, i.e., the distance from the backward-facing step to the reattachment point, are shown for all cases in Table II. As we increase the tank pressure, the separation zone length becomes longer. The separation zone length of a round needle end is smaller than that of a square needle end. Therefore proper changes in the nozzle shape can reduce separation zone length and thus pressure losses by expanding low supersonic flow smoothly after the nozzle throat.
Conclusions
We have studied axisymmetric transonic/supersonic flow fields in the main nozzle of an air-jet-loom using the compressible, upwind flux, difference-splitting Navier-Stokes method. At air tank pressures of 4 kg^sub f^/ cm^sup 2^ or higher, a choking phenomenon occurs, not only at the nozzle throat but also at the acceleration tube exit. Flow velocity distribution near the main nozzle throat and at the acceleration tube exit is similar at air tank pressures of 4 kg^sub f^/cm^sup 2^ or higher. Due to flow choking at the tube exit near air pressures of 4 kg^sub f^/ cm^sup 2^, the optimum air tank pressure is about 4 kg^sub f^/cm^sup 2^ in the main nozzle system we have considered. Even though the change in acceleration tube length does not change flow characteristics much, the tube length seems to be related more to weft stability and thrust force. The change in nozzle shape from the square shaped needle end to the circular arc causes static pressure to rise in the tube and reduces total pressure losses in the nozzle, i.e., we can reduce air consumption slightly.
ACKNOWLEDGMENT
This work was partly supported by the Brain Korea 21 project.
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Manuscript received June 26, 2000; accepted December 12, 2000.
T. H. OH, S. D. KIM AND D. J. SONG
School of Mechanical Engineering, Yeungram University, Gyongsan 712-749, South Korea
