Study on three-dimensional expansion characteristics of four wall combustion-gas jets in confined liquid space

Transcription

1 Acta Mech. Sin. (2017) 33(2): DOI /s RESEARCH PAPER Study on three-dimensional expansion characteristics of four wall combustion-gas jets in confined liquid space Zhitao Hu 1 Yonggang Yu 1 Received: 25 September 2016 / Revised: 9 January 2017 / Accepted: 17 January 2017 / Published online: 7 March 2017 The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017 Abstract To explore further the launch mechanism of the new underwater launching technology proposed in this paper, the expansion characteristics of four wall combustion-gas jets in confined liquid space must be studied firstly. The experimental device is designed, and the high-speed digital photographic system is adopted to obtain the expansion sequence processes of Taylor cavities formed by the four wall jets. Meanwhile, the influence of the injection pressure on the axial expansion property of the four wall jets is discussed. Based on the experiments, a three-dimensional unsteady mathematical model is established to simulate the turbulent flow process of the four wall jets expanding in liquid, and the temporal and spatial distribution laws of phase, pressure, temperature, and velocity and the evolution rules of vortices are illustrated in detail. Results show that, accompanied by the jets expanding downstream, the four wall combustion-gas jets get close to each other and achieve convergence eventually under induction of the interference effect between multiple jets. Meanwhile, the heads of the Taylor cavities separate from the observation chamber wall and offset to the central axis of the observation chamber with time going on. The numerical simulation results of the four wall combustion-gas jets coincide well with the experimental data. Keywords Wall jet Taylor cavity Kelvin-Helmholtz instability Interference effect B Yonggang Yu yygnjust801@163.com 1 School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing , China 1 Introduction When a gaseous jet is injected into water through a nozzle, it forms a complex multiphase system, and the flow structure and process are essentially unsteady and turbulent. This process can be found in a variety of engineering applications, such as metallurgy [1], nuclear [2], turbine [3], and underwater launching [4]. During the launching process of a submerged underwater-gun, a large amount of water exists in front of the projectile, which leads to sudden changes of the ballistic characteristics, sharp increases in the chamber pressure, and accidents of breech blow. Therefore, this paper proposes a new underwater launching technology for a submerged underwater-gun, that is, propellant gases are guided and then discharged through the grooves in the gunbarrel sidewall in front of the projectile accompanied by draining off the water in the gunbarrel. Thus, with the movement of the projectile, it can effectively reduce the resistance at the front-end of the projectile in the launching process under this technology. To explore further the launch mechanism of the new underwater launching technology, the expansion characteristics of four wall combustion-gas jets in confined liquid space, which belongs to the submerged gaseous jet category, must be studied firstly. Based on the method used, the research on submerged gaseous jets can be categorized as experimental and numerical. Experimental research and observation are described as follows. Experiments by Loth and Faeth [5,6] were carried out on the high-pressure gaseous jets submerged in water to measure the centerline pressure distributions and timeaveraged void fractions. The work by Dai et al. [7] and Shi et al. [8] made visualized experimental research on the process of high pressure air jets expanding freely under water and large pressure pulsations related to the phenomenon

2 342 Z. Hu, Y. Yu of shock waves feedback were elucidated by performing detailed measurements on jet pressure distribution. The process of a gaseous jet in water was measured by Arghode and Gupta [9] with different exit nozzle cross-sections, and the laws of pressure fluctuation and the conversion mechanism between bubbling regime and jetting regime were revealed. In their paper, Weiland and Vlachos [10] performed direct measurements of the interfacial behavior of water-submerged high speed gas jets with Mach numbers ranging from subsonic to supersonic by using high speed digital photography and indicated that the jets have a preferential pinch-off position which corresponds to the location of the maximum streamwise-velocity turbulence fluctuations. An experimental investigation was made by Voropayev et al. [11] on the evolution of turbulent jets into a cylinder by using digital video recordings and particle image velocimetry (PIV) to map the flow structures, and the frequency of jet switching, mean flow, and turbulence characteristics were inferred. The work of Yu et al. [12] investigated the flow structure and turbulent mixing of pulsed jets issuing from a circular nozzle by using acetone planar laser-induced fluorescence (PLIF) and interpreted different features of gaseous jets by monitoring axial and various radial cross-sections under different injection pressure conditions. Comprehensive experimental studies were conducted by Law and Herlina [13,14] on kinematic and scalar mixing characteristics of a turbulent circular wall jet with the nonintrusive laser imaging approach of combining PIV and PLIF, and the results show that the growth of the lateral length scale is five times that of the normal length scale. Work by Agelin and Tachie [15] investigated the three-dimensional turbulent wall jets by using a PIV technique with three Reynolds numbers of 5000, 10,000, 20,000, which is much higher than that in previous experiments, and the salient features of the wall jets, such as mean velocities, turbulence intensities, and Reynolds shear stresses were analyzed. Numerically, Tang et al. [16,17] revealed the phenomena of bulge, necking/breaking, and back-attack in the jets expansion processes by numerically simulating the process of jet expanding under water. A mathematical model was proposed by Cheng and Liu [18] for an exhaust gas bubbles in missiles launched in an underwater process through using different numerical methods to describe the gas flow field and water flow field, respectively. In their work, Zhang et al. [19] performed a comprehensive investigation on the turbulent circular wall jet including both the flow and mixing characteristics using the large eddy simulations (LES) approach with proper near-wall model. Cold-state experiments were carried out by Yu et al. [20,21] to investigate the expansion process and gas-liquid mixing characteristic of a single gas jet in a stepped-wall observation chamber, and they made numerical simulations to validate the experimental results. Work by Xue et al. [22,23] developed the experimental and numerical studies on twin and multiple gas jets in cylindrical stepped-wall filling liquid chamber and gave detailed pressure, density, temperature, and velocity contours in flow field. To understand deeply the interaction between gas and liquid in cavitating flow, Li et al. [24] and Wei et al. [25] developed different numerical models to compute the transport processes of the cavitating flow. Based on the previous studies, this paper carries out experimental study on the expansion characteristics of four wall combustion-gas jets in cylindrical observation chamber, and then the influences of the injection pressure on the axial expansion property are discussed. Furthermore, a three-dimensional unsteady mathematical model is established to provide detailed flow field information including phase, pressure, temperature, and velocity. Finally, this paper provides a valuable theoretical basis for future studies on the launch mechanism of a submerged underwater gun. 2 Experimental device and principle As shown in Fig. 1, the experimental device is composed of a high-pressure combustor, a connector, a nozzle with four wall orifices distributing uniformly along the circumferential Fig. 1 Sketch of the experimental device. 1 Ignition electrode, 2 deflagrating gunpowder, 3 high-pressure combustor, 4 copper sealing film, 5 connector, 6 nozzle, 7 narrow wall orifice, 8 cylindrical observation chamber

3 Study on three-dimensional expansion characteristics of four wall combustion-gas jets 343 Fig. 2 Schematic of the measurement system direction, and an observation chamber. As can be seen from the figure, the narrow wall orifices are presented on the M- M sectional view of experimental device and are rectangular shape with 3 mm 4 mm in structure size. The observation chamber is a cylindrical structure of Φ55 mm 136 mm, made of transparent organic glass to facilitate observation, and filled with liquid medium. The working principle is that the deflagrating gunpowder filled in the high-pressure combustor is ignited by the pulse electric ignition system to generate combustion gases with high temperature and high pressure. Then the combustion gases are injected into the transparent cylindrical observation chamber through four wall orifices when the injection pressure gets to the rupturing threshold value of the copper sealing film, and the four wall combustion-gas jets are formed in liquid. In order to reduce the effects of gravity on the combustion-gas jets, the device is placed vertically upward, that is, the combustion gases with high temperature and high pressure are injected from downward to upward, and the outlet of the observation chamber is opened and directly connects with the air. Figure 2 is a schematic of the measurement system. Two plane mirrors placed at an angle of 45 are used to acquire both the front view and the side view of the expansion process of the four wall jets in liquid, which are recorded by means of a high-speed digital photographic system by simultaneously focusing on the virtual images in each plane mirrors. In addition, the pulse electric ignition system and the high-speed digital photographic system are synchronously controlled in order to obtain the complete expansion process, and the experimental device and the high-speed digital photographic system are reasonably placed under sunny condition to achieve the clear high-speed images of the jet expansion process. The filming frequency of the high-speed digital photographic system is 4000 fps in experiments. 3 Experimental results and discussions 3.1 The expansion sequence process of four wall combustion-gas jets A typical experimental condition is chosen as an example to illustrate the expansion characteristics of the four wall combustion-gas jets, which is as follows: the injection pressure is 20 MPa controlled by adjusting the mass of the powder charge and the thickness of copper sealing film, the nozzle shape is rectangular, and the nozzle size is 3 mm 4mm. Figures 3 and 4 show the sequence maps of the four wall combustion-gas jets expanding in a cylindrical observation chamber. According to the shape change of Taylor cavities at different times, the whole expansion process is divided into two periods: the period of independent expansion, in which four Taylor cavities expand forward independently, and the period of joint expansion, in which four Taylor cavities achieve convergence and expand forward jointly. Figure 3 shows the period of independent expansion. As can be seen from the figure, the combustion gases rapidly expand into bubbles showing obviously mutual independence, and due to the Taylor instability, the heads present in a zigzag, when they are just injected into liquid medium at t = 0.25 ms. At t = 1 ms, as a result of intense turbulent mixing and interference effect, the outer contours of the Taylor cavities are irregular and become more obvious with time going on. At t = 1.75 ms, the period of independent expansion terminates when the four wall jets meet each other. Meanwhile, the heads of the Taylor cavities gradually separate from the wall and offset to the central axis of the observation chamber under induction of the interference effect between multiple wall jets. Figure 4 shows the sequence maps of the period of joint expansion. As can be seen from Fig. 4a, the four wall jets start to converge at the bottom of Taylor cavity and then

4 344 Z. Hu, Y. Yu a b t=0.25 ms t=0.5 ms t=0.75 ms t=1 ms t=1.25 ms t=1.5 ms t=1.75 ms t=0.25 ms t=0.5 ms t=0.75 ms t=1 ms t=1.25 ms t=1.5 ms t=1.75 ms Fig. 3 The period of independent expansion in expansion process. a Side-view. b Front-view expand downstream jointly with an increase in convergence region. Meanwhile, from Fig. 4b, the heads of Taylor cavities are fan-shaped within t = ms and then transform into conical after 3 ms. The reason for this type of phenomenon is that the radial expansion velocities of the four wall jets are great compared to the axial velocities due to turbulent mixing and interference effect within t = ms, while the axial expansion velocities of the four wall jets play a dominant role after t = 3 ms because the four Taylor cavities are too close, resulting in a restraint on the radial expansion velocities. From the whole expansion process, it can be seen that the four wall jets are symmetric in general. 3.2 The influence of the injection pressure To analyze the influence of the injection pressure on the axial expansion property of the four wall jets, experiments under three different injection pressures, that is 12, 20, 28 MPa respectively, are performed, and all the nozzle structures are rectangular and 3 mm 4 mm in size. The axial expansion displacement is determined by taking the average of the instant displacement of the front-end interface of the four Taylor cavities at different times by dealing with the sequence maps. Figure 5 shows the axial expansion displacement curves of the four wall jets under different injection pressures. Repeated tests under the three conditions have been conducted in experiment, and the error bars are plotted in the figure, from which we can see that the maximum axial displacement error is less than 5 %. Furthermore, as can be seen from the figure, as the injection pressure increases, the power source of the combustion-gas jets enhances, which leads to the heads of the four wall jets reaching to the exit of the cylindrical chamber earlier. Before the injection pressure reaches 20 MPa, the axial displacement of the four wall jets increases substantially, and with a further increase of the injection pressure to 28 MPa, the axial displacement increase little. Because the axial displacement of the four wall jets is mainly decided by the nonlinear coupling effect of inlet axial turbulent kinetic energy and turbulent mixing. For the case of injection pressure of 12 MPa, the inlet axial turbulent kinetic energy is the least, but the decay rate of axial turbulent kinetic energy is also the slowest due to the lowest turbulent mixing intensity with resulting from the weakest Kelvin-Helmholtz instability effect. For the case of 20 MPa injection pressure, axial turbulent kinetic energy at the inlet increases with increasing injection pressure, which leads to the enhancement of Kelvin-Helmholtz instability effect, but the axial turbulent kinetic energy decays faster as a result of more intense turbulent mixing. For the case of 28 MPa injection pressure, the inlet axial turbulent kinetic energy is the largest, but the decay rate of axial turbulent kinetic energy is also the fastest due to the highest turbulent mixing intensity resulting from the strongest Kelvin-Helmholtz instability

5 Study on three-dimensional expansion characteristics of four wall combustion-gas jets 345 a b t=2 ms t=2.5 ms t=3 ms t=3.5 ms t=4 ms t=5 ms t=6 ms t=2 ms t=2.5 ms t=3 ms t=3.5 ms t=4 ms t=5 ms t=6 ms Fig. 4 The period of joint expansion in expansion process. a Side-view. b Front-view Table 1 The fitting parameters for the axial displacement-time curves of Taylor cavities Injection pressure (MPa) R 2 B 0 (mm) B 1 (mm) B 2 (ms) Fig. 5 The relation of x t under different injection pressures effect. In summary, as the injection pressure increases, the axial displacement of the four wall jets increases, but presents nonlinear variation. After the injection pressure reaches 20 MPa, the turbulent mixing effect plays a dominant role and the increment of the axial displacement of the four wall jets decreases with the injection pressure increasing further. Thus, the axial displacement has little increase when the injection pressure increases to 28 MPa. A first-order exponential decay equation is chosen to fit the average axial displacement-time curves of Taylor cavities based on the data obtained from sequence images. x(t) = B 0 + B 1 e t/b 2, where x(t) is the average axial displacement of Taylor cavities and the unit is mm, t is the expansion time with the unit of ms, and B 0, B 1, and B 2 are fitting parameters and the units are mm, mm, and ms, respectively. As shown in Table 1, R 2 is the correction coefficient. 4 Mathematical and physical models 4.1 Physical models According to the experimental study on the expansion process of the four wall combustion-gas jets in liquid medium, simplified assumptions of the physical process are made as follows (1) The expansion process of the four wall combustion-gas jets in liquid is a three-dimensional unsteady process, the volume of fluid (VOF) model is used to calculate the

6 346 Z. Hu, Y. Yu two-phase flow, and the Realizable k ε model is used to calculate the turbulent mixing. (2) Because the whole expansion process of the four wall combustion-gas jets in liquid is very short, namely less than 8 ms, secondary influence factors such as the phase transitions and the gravity of the combustion gases are ignored in this paper. (3) The combustion gas jets are approximated as compressible and ideal gas jets. 4.2 Mathematical model The continuity, momentum, energy conservation, state, and turbulent equations for unsteady gas jets flow are given as follows (1) Continuity equation ( ) α q ρ q + (α q ρ q v ) = 0. (1) t This model has two phases: the gas phase and the liquid phase. When q = 1, α 1 is the volume fraction of the gas phase, and when q = 2, α 2 is the volume fraction of the liquid phase, and α 1 + α 2 = 1. (2) Momentum conservation equation ( (ρv)+ (ρvv) = p+ [ μ v + v T)], (2) t where, the density and viscosity of two-phase mixture are given as ρ = α 1 ρ 1 + α 2 ρ 2 and μ = α 1 μ 1 + α 2 μ 2. (3) Energy conservation equation t (ρ E) + [v (ρ E + p)] = (k eff T ). (3) In the VOF model, the energy, E, and the temperature, T, are treated as mass-averaged variables as follows E = 2 α q ρ q, 2 α q ρ q E q/ q=1 q=1 2 T = α q ρ q, q=1 α q ρ q T q/ 2 q=1 where, E q (total energy for q-th phase) is based on the specific heat of that phase and the shared temperature, k eff is effective thermal conductivity. (4) State equation p = ρ RT. (4) (5) Turbulent flow equation The Realizable k ε model is used to simulate the turbulent mixing in this paper. The turbulence kinetic energy, k, and its dissipation rate, ε, are obtained from the following transport equations t (ρk) + (ρku i ) = x i t (ρε) + x i (ρεu i ) = [( μ + μ ) ] t k x j σ k x j + G k ρε Y M, (5) [( μ + μ ) ] t ε x j σ ε x j ε 2 + ρc 1 Sε ρc 2 k + νε, (6) where μ t is the turbulent viscosity and can be calculated as follows μ t = ρc μ k 2 /ε. C μ is a variable related to strain rate and can be computed from C μ = 1 A 0 + A S ku /ε. In these equations, G k represents the generation of turbulence kinetic energy due to the mean velocity gradients. Y M represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. δ k and δ ε are the turbulent Prandtl numbers for k and ε, respectively. 4.3 Computational domain and grid Figure 6 shows the simulation domain configured based on the experimental setup. The Cartesian coordinate system is employed, where x, y, and z represents the distance in streamwise, normal and spanwise direction, respectively, and u, v, and w represent velocities that correspond to x, y, and z, respectively. Considering the symmetry of the flow field, a quarter of the flow field is selected for calculation. There are four boundary conditions, which are, respectively, the pressure inlet boundary, the pressure outlet boundary, the wall, and the symmetry planes. The mesh was a structured type with a total of 610,000 cells with a higher resolution near the inlet, near wall region and the interior of nozzle tube. Furthermore, the grid convergence had been tested for the estimation of uncertainty due to the discretization in computational fluid dynamics (CFD) simulations with a more refined grid of 900,000 cells, and the comparison plots of the axial displacement of numerical results under different grid resolutions can be seen in Fig. 7, which shows that the difference is little with an estimated maximum error of 5 %.

7 Study on three-dimensional expansion characteristics of four wall combustion-gas jets 347 Fig. 6 Schematic diagram of the computational domain Fig. 7 The grid convergence test 4.4 Initial and boundary conditions At initial time, the four wall combustion-gas jets have not been exhausted from the nozzle, so the initial conditions are as follows: the observation chamber is filled with water, the nozzle tube is filled with air, and the fluid is stationary with a homogenous temperature of 300 K and pressure of 1 atm. The pressure inlet boundary conditions are determined by the experiment: the injection pressure is 20 MPa and the temperature is about 2200 K. The gas volume fraction is specified at the pressure inlet, α 1 = 1, indicating that the injected fluid is purely gas. The outlet of cylindrical chamber is directly connected with the air, so the pressure outlet boundary conditions are the atmospheric parameters: p = kpa, T = 300 K. The wall adopts no-slip heat insulation condition. 5 Numerical simulations and discussions Based on the experiment, the numerical conditions are as follows: the injection pressure is 20 MPa and the nozzle structure size is a 3 mm 4 mm rectangle. The numerical simulation of the expansion process of the four wall combustion-gas jets in liquid is conducted by means of CFD code, among which the VOF method is adopted for simulating the multiphase flow and tracking the free surface of two phases, the Realizable k ε model with enhanced wall functions is applied to calculating the turbulence mixing between combustion gases and liquid. In the present numerical calculation, the mesh near the wall region has been further refined so that the y + value is about 1. In addition, a second-order upwind scheme is applied for the momentum and energy equation, the geometric reconstruction scheme is applied for the continuity equation. The convergence criteria are: the residuals of energy equation are less than 10 6, others are less than 10 3, and the time step size is set as s so that the global Courant number is less than Since the parameters in the flow field change largely, the distribution of the infield of nozzle tube is omitted in the following analyses to better illustrate the characteristics of parameters in the flow field. 5.1 The phase distribution Figure 8 shows the three-dimensional phase distribution of the four wall jets in the cylindrical observation chamber. As can be seen from the figure, when the four wall jets just inject into the liquid medium, namely at t = 1 ms, the four wall jets are obviously separate, and the heads present an irregular shape as serration, and the outer contours are relatively regular with a few folds. As time goes on, the four wall jets get close to each other at the same time with the heads progressively separating from the wall due to the interference effect and the outer contours become irregular. At t = 3.5 ms, the four wall jets achieve convergence with a distinct radial expansion zone at the bottom of the Taylor cavity and reach full convergence at t = 5 ms with some water left between the four wall jets. From the figure, we can

8 348 Z. Hu, Y. Yu a b t=1 ms t=2 ms t=3 ms t=3.5 ms t=4 ms t=5 ms t=6 ms t=1 ms t=2 ms t=3 ms t=3.5 ms t=4 ms t=5 ms t=6 ms Fig. 8 The three-dimensional phase distribution of the four wall jets. a Side-view. b Front-view Fig. 9 The expansion displacement of the four wall jets. a Comparison of axial displacement. b Simulation of radial displacement observe significant turbulent folds on the interfaces and parts of the droplets in the Taylor cavities, which just reflect the turbulent mixing degree and the intensity of Kelvin-Helmholtz instability due to large tangential velocity difference between the gases and the liquid. Combined with Figs. 3 and 4, the leading features of the four wall combustion-gas jets expanding in liquid medium revealed in numerical simulation mainly coincide well with the experimental results. The comparison curves on the axial displacement of the Taylor cavity heads are obtained by dealing with the experimental processes in Figs. 3 and 4, and the three-dimensional phase distribution figures in Fig. 8, as can be seen in Fig. 9a. From the figure, the maximum error between the experimental data and simulation results is 8 %, while the difference at most places is small, which shows that the numerical simulation results coincide well with the experimental data. Figure 9b shows the radial displacement-time curve in the

9 Study on three-dimensional expansion characteristics of four wall combustion-gas jets 349 Fig. 10 The comparison of the axial displacement for the cases of 12 and 28 MPa. a p = 12 MPa. b p = 28 MPa t=1 ms t=2 ms t=3 ms t=4 ms t=5 ms t=6 ms Fig. 11 The phase distribution on x = 5 mm atdifferent time numerical simulation. The radial expansion displacement of the four wall jets increases with time going on, and reaches to converge at t = 3.5 ms. Figure 10 shows the comparison plots of the axial displacement of experiment and simulation for the cases of 12 and 28 MPa injection pressures. As can be seen from the figure, the numerical simulation results quantitatively accord with the experimental data and the differences between experimental data and numerical results under the conditions are all less than 8 %, which is similar to the case of 20 MPa injection pressure. Figure 11 shows the phase distribution of the four wall jets in the cylindrical observation chamber on the cross-section of x = 5 mm, wherein the black represents the combustiongas jets zone and the gray represents the liquid medium zone. As can be seen from the figure, at t = 1 ms, the four wall jets just inject into the liquid and present mutual independence, wherein the combustion-gas jets zone presents semi-circle structure which reveals the mechanism of axis switching in rectangular jets. As the four wall jets expand downstream, they move closer and closer and achieve convergence eventually under induction of the interference effect between multiple jets. Meanwhile, the gas liquid interface folds significantly due to the intense mixing effect and even some droplets are absorbed into Taylor cavities through the entrainment effect. 5.2 The characteristics of the flow field between two opposite wall jets According to Fig. 6, there are two relationships between the four wall combustion-gas jets, that is, opposite and adjacent. The following discussions are focused on two planes: the plane A (at z = 0) and the plane B (at z = y) which illustrate two opposite wall jets and two adjacent wall jets spreading along the flow direction, respectively The vortex distribution in plane A Figure 12 shows the evolution law of vortices in the expansion process of the four wall combustion-gas jets in plane A. As can be seen from the figure, there is a pair of backflow vortices in the Taylor cavities due to high velocity difference between the combustion gases and the stationary liquid, which move downstream and close to each other over time with an increase in the vortex scale. After t = 4 ms, many small-scale vortices are formed on the interfaces of Taylor cavities with confused and disordered distribution, which results in intense turbulent mixing and entrainment effect between combustion gases and liquid medium, and the intensity of the turbulent mixing would be enhanced further with the vortices forming and moving in the flow field.

10 350 Z. Hu, Y. Yu Fig. 12 The vortex distribution of the four wall jets in plane A Fig. 13 (Color online) The static pressure distribution of the four wall jets in plane A The static pressure distribution in plane A Figure 13 shows the static pressure distribution of the four wall combustion-gas jets in plane A. As can be seen from the figure, when the high-speed combustion-gas jet injects into Taylor cavity and impacts on the liquid surface in the position where the Taylor cavity just separates from the observation chamber wall, two high-pressure regions are formed due to the pressure oscillation near the gas liquid interface. The pressure values of the high-pressure regions decrease gradually as time goes on. The pressure waves in the liquid medium propagate downstream and transform from the plane waves into a hump shape due to the interference effect. Meanwhile, it can be observed that the mutual interval structure of high pressure and low pressure appears near the nozzle beginning at t = 2 ms showing large pressure fluctuation in the flow field. To analyze further the characteristics of pressure, Fig. 14 shows the static pressure curves in different range of time Fig. 14 The static pressure distribution along the centerline of nozzle along the centerline of one nozzle. As can be seen from the figure, the distribution trends of static pressure under different times are similar, all decreasing to low peak and then

11 Study on three-dimensional expansion characteristics of four wall combustion-gas jets 351 Fig. 15 (Color online) The temperature distribution of the four wall jets in plane A increasing to high peak dramatically. The reason caused this kind of phenomenon is: the static pressure at the nozzle exit is 2.4 MPa and is much higher than that of the ambient liquid which leads to an immediate expansion of the combustiongas jets for equilibrating with the ambient pressure. Then expansion waves are formed in the Taylor cavities, which are reflected on the jet boundaries, and compression waves are formed. When the combustion-gas jets pass through the expansion waves, the pressure decreases to low peak and then dramatically increases to a high peak after they pass through the compression waves. The position of low peak, as well as high peak of the pressure curve moves downstream over time with a decrease in the value. Meanwhile, there are two high peaks formed on the downstream region of the flow field, because the high-speed combustion-gas jets impact the liquid surface in the position where the Taylor cavities just separate from the observation chamber wall which leads to the pressure oscillation near the gas-liquid interface The temperature distribution in plane A Figure 15 shows the temperature distribution of the four wall combustion-gas jets in plane A. As can be seen from the figure, at t = 1 ms, combustion-gas jets with high temperature are compressed by surrounding quiescent liquid medium, which leads to two high-temperature regions formed at the heads of the Taylor cavities disappearing with time due to intense turbulent mixing between combustion gases and liquid. Meanwhile, there is another pair of high-temperature regions at the bottom of the observation chamber caused by the reflow vortices. In addition, as the four wall jets expand downstream, the affected regions of temperature increase along axial and radial directions, of which the heads offset to the central axis of the cylindrical observation chamber gradually. Fig. 16 The temperature distribution on the centerline of nozzle To analyze further the characteristics of temperature, Fig. 16 shows the temperature curves in different range of time along the centerline of one nozzle. From above, we know there are expansion waves and compression waves in the near field of nozzle. The combustion-gas jets expand immediately when exhausted out of the nozzle, and the temperature sharply decreases to low peak. The changing trend of low peak of the temperature curve is similar to that of the static pressure, presenting moving downstream with an increase in temperature value. The temperature dramatically increases to high peak when the combustion-gas jets pass through the compression waves. The high peak value of temperature curve increases over time at first and reaches to the maximum of 1655 K at t = 4 ms, and then decreases to 1613 K at t = 5 ms and 1462 K at t = 6ms The velocity distribution in plane A Figure 17 shows the velocity distribution of the four wall combustion-gas jets in plane A. As can be seen from the fig-

12 352 Z. Hu, Y. Yu Fig. 17 (Color online) The velocity distribution of the four wall jets in plane A Fig. 18 Non-dimensional profiles of streamwise velocity in plane A. a On different sections. b At different times ure, the combustion-gas jets expand rapidly and transform into supersonic jets when just exhausted out of the nozzle. Then the velocities decrease sharply after the combustiongas jets pass through the compression waves. Meanwhile, the regions with lower velocity offset to the central axis of the cylindrical observation chamber, while the regions with higher velocity always locate at the cores of the wall jets. Figure 18 shows the non-dimensional profiles of streamwise velocity along y-direction in plane A, where D is the hydraulic diameter of nozzle that is D = 3.4 mm, u m is local maximum streamwise velocity, y m/2 is the vertical distance from the position of u = u m /2tox/mm axis in plane A. Figure 18a shows the non-dimensional profiles of streamwise velocity on different x/d sections. As can be seen, all the streamwise velocity profiles except x/d = 26 collapse y into a single curve within the limits of y m/2 < 2, and the streamwise velocity profiles show irregular and even to negative value when y y m/2 > 2, which reveals that the central flow field of the observation chamber is very complicated due to the intense interference effect, but the streamwise velocity profiles in near wall region are self-similar. Figure 18b shows the non-dimensional profiles of streamwise velocity at different times. As can be seen, the velocity profiles all y collapse into a single curve within the limits of y m/2 < 2, but the coincidence degree of streamwise velocity profiles is y very weak within y m/2 > 2. It reveals that the streamwise velocity profiles show self-similarity in near wall region and fluctuation in the central field of the observation chamber. 5.3 The characteristics of the flow field between two adjacent wall jets As shown in the following figures, l is the vertical distance to x/mm-axis in plane B The static pressure distribution in plane B Figure 19 shows the static pressure distribution of the four wall combustion-gas jets in plane B. As can be seen from the figure, the leading features of the static pressure distribution in plane B, such as the high-pressure regions and the mutual

13 Study on three-dimensional expansion characteristics of four wall combustion-gas jets 353 Fig. 19 (Color online) The static pressure distribution of the four wall jets in plane B Fig. 20 (Color online) The temperature distribution of the four wall jets in plane B interval structure of high pressure and low pressure, are similar to those in plane A. Meanwhile, it can be observed that parts of small zones with high pressure identified in the rectangular frame are formed on the dowmstream region of the flow field, as the high-speed combustion gases impact on the liquid droplets that are entrained into Taylor cavities The temperature distribution in plane B Figure 20 shows the temperature distribution of the four wall combustion-gas jets in plane B. As can be seen from the figure, the affected regions of temperature spread along axial and redial direction with the jets expanding downstream, and achieve convergence presenting bowknot shape at 6 ms. Meanwhile, a high-temperature zone appears in middle of the bottom wall at t = 5 ms with an increase in the area over time. The above phenomena are because the interval distance between two adjacent wall jets is so small that the jets could achieve full convergence The velocity distribution in plane B Figure 21 shows the velocity distribution of the four wall combustion-gas jets in plane B. As can be seen from the figure, the regions with higher velocity in plane B are larger than those in plane A, and the regions with lower velocity in plane B offset to the central axis of the cylindrical observation chamber more obviously than that in plane A, which shows that the interference effect between two adjacent wall jets is more intense than that between two opposite wall jets. To analyze further the characteristics of the velocity in plane B, Fig. 22 shows the non-dimensional profiles of streamwise velocity in plane B. Where l m/2 is the vertical distance from the poison of u = u m /2 to x/mm axis in plane B. As can be seen from Fig. 22a, the streamwise velocity profiles on different x/d sections can not overlap together, which reveals that the streamwise velocity profiles do not present spatial self-similar property in plane B because the four wall jets achieve convergence so quickly that leads to intense inter-

14 354 Z. Hu, Y. Yu Fig. 21 (Color online) The velocity distribution of the four wall jets in plane B Fig. 22 Non-dimensional profiles of streamwise velocity in plane B. a On different sections. b At different times ference effect. As can be seen from Fig. 22b, the velocity profiles at any time collapses into a single curve within the limits of < 2, but present dishevelled distribution within l l m/2 l l m/2 > 2. It is obvious that the streamwise velocity distribution presents temporal self-similarity property in near wall region and oscillation characteristic in central flow field of the observation chamber. 6 Conclusions Based on the experimental and numerical results, the expansion characteristics of the four wall combustion-gas jets in cylindrical observation chamber are summarized as follows (1) In the expansion process of the four wall combustiongas jets in confined liquid medium, the four wall jets get close to each other and achieve convergence eventually under induction of the interference effect between multiple wall jets. Meanwhile, the heads of the Taylor cavities separate from the observation chamber wall and offset to the central axis of the observation chamber with time going on. (2) The axial expansion property of the four wall jets is relevant with the injection pressure, namely, the axial expansion displacement is rising as the injection pressure increases. The relationship of the axial displacementtime curve satisfies empirical formula, that is, x(t) = B 0 + B 1 e t/b 2. (3) It is apparent that the mutual interval structure of high pressure and low pressure can be observed in the near field of nozzle. The reason caused this kind of phenomenon is: the static pressure at the nozzle exit is 2.4 MPa and is much higher than that of the ambient liquid, which leads to fast expansion of the combustiongas jets for equilibrating with the ambient pressure. Then expansion waves are formed in the Taylor cavities, which are reflected on the jet boundaries, and compression waves are formed. When the combustion-gas jets pass through the expansion waves, the pressure decreases to low peak and then dramatically increases to high peak after the combustion-gas jets pass through the compression waves.

15 Study on three-dimensional expansion characteristics of four wall combustion-gas jets 355 (4) As the appearance of the expansion waves and the compression waves, the temperature oscillate dramatically near the nozzle. Two types of high-temperature region are formed in the flow field: one is the head region with high-temperature caused by compression effect from surrounding quiescent liquid medium at the head of Taylor cavity, which disappears over time due to violent turbulent mixing between combustion gases and liquid, and the other is the bottom region with high-temperature caused by the reflow vortices. Moreover, the affected regions of temperature in plane B are larger than those in plane A, and achieve convergence presenting bowknot shape at 6 ms. (5) The combustion-gas jets expand rapidly and transform into supersonic jets, and then the velocities decrease sharply after the combustion-gas jets pass through the compression waves. Furthermore, the non-dimensional profiles of streamwise velocity near wall region present temporal and spatial self-similar property in plane A, but only present temporal self-similar property in plane B. This is because the interval distance between two adjacent wall jets is so small that the interference effect between two adjacent wall jets is more intense than that between two opposite wall jets. (6) The values of the numerical axial expansion displacement of the four wall combustion-gas jets are in good agreement with the experimental data. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant ). References 1. Nguyen, A.V., Evans, G.M.: Computational fluid dynamics modeling of gas jets impinging onto liquid pools. Appl. Math. Model. 30, (2006) 2. Darmana, D., Deen, N.G., Kuipers, J.A.M.: Detailed modeling of hydrodynamics, mass transfer and chemical reactions in a bubble column using a discrete bubble model. Chem. Eng. Sci. 60, (2005) 3. Chaudhari, M., Puranik, B., Agrawal, A.: Heat tranfer characteristics of synthetic jet impingement cooling. Int. J. Heat Mass Transf. 53, (2010) 4. Lindau, J., Venkateswaran, S., Kunz, R., et al.: Multiphase computations for underwater propulsive flows. In: 16th AIAA Computational Fluid Dynamics Conference, Orlando, Florida, AIAA (2003) 5. Loth, E., Faeth, G.M.: Structure of underexpanded round air jets submerged in water. Int. J. Multiph. Flow 15, (1989) 6. Loth, E., Faeth, G.M.: Structure of plane underexpanded air jets into water. Am. Inst. Chem. Eng. 36, (1990) 7. Dai, Z.Q., Wang, B.Y., Qi, L.X., et al.: Experimental study on hydrodynamic behaviors of high-speed gas jets in still water. Acta Mech. Sin. 22, (2006) 8. Shi, H.H., Wang, B.Y., Dai, Z.Q.: Research on the mechanics of underwater supersonic gas jets. Sci. China Phys. Mech. Astron. 53, (2010) 9. Arghode, V.K., Gupta, A.K.: Jet characteristics from a submerged combustion system. Appl. Energy 89, (2012) 10. Weiland, C., Vlachos, P.P.: Round gas jets submerged in water. Int. J. Multiph. Flow 48, (2013) 11. Voropayev, S.I., Sanchez, X., Nath, C., et al.: Evolution of a confined turbulent jet in a long cylindrical cavity: homogeneous fluids. Phys. Fluids 23, (2011) 12. Yu, J.Z., Vuorinen, V., Kaario, O., et al.: Visualization and analysis of the characteristics of transitional underexpanded jets. Int. J. Heat Fluid Flow 44, (2013) 13. Law, A.W.K., Herlina, H.: An experimental study on turbulent circular wall jets. J. Hydraul. Eng. 128, (2002) 14. Law, A.W.K., Herlina, : Measurements of turbulent mass transport of a circular wall jet. Int. J. Heat Mass Transf. 45, (2002) 15. Agelin, C.M., Tachie, M.F.: Characteristics of turbulent threedimensional wall jets. J. Fluids Eng. Trans. ASME 133, (2011) 16. Tang, J.N., Wang, N.F., Shyy, W.: Flow structures of gaseous jets injected into water for underwater propulsion. Acta Mech. Sin. 27, (2011) 17. Tang, J.N., Tseng, C.C., Wang, N.F.: Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents. Acta Mech. Sin. 28, (2012) 18. Cheng, Y.S., Liu, H.: A coupling model of water flows and gas flows in exhausted gas bubble on missile launched underwater. J. Hydrodyn. 19, (2007) 19. Zhang, S., Law, A.W.K., Zhao, B.: Large eddy simulations of turbulentcircularwalljets.int.j.heatmasstransf.80, (2015) 20. Yu, Y.G., Chang, X.X., Zhao, N., et al.: Study of bulk-loaded liquid propellant combustion propulsion processes with stepped-wall combustion chamber. J. Appl. Mech. 78, (2011) 21. Yu, Y.G., Zhang, Q., Zhao, N., et al.: Experimental study and numerical simulation on propagation properties of a plasma jet in a cylindrical liquid chamber. J. Appl. Mech. 80, (2013) 22. Xue, X.C., Yu, Y.G., Zhang, Q.: Study on the influences of interaction behaviors between multiple combustion-gas jets on expansion characteristics of Taylor cavities. Acta Mech. Sin. 31, (2015) 23. Xue, X.C., Yu, Y.G., Mang, S.S.: Physical characteristics on high-pressure combustion and propelling process of bulk-loaded energetic liquid. Appl. Therm. Eng. 98, (2016) 24. Li, X.B., Wang, G.Y., Yu, Z.Y., et al.: Multiphase fluid dynamics and transport processes of low capillary number cavitating flows. Acta Mech. Sin. 25, (2009) 25. Wei, Y.J., Tseng, C.C., Wang, G.Y.: Turbulence and cavitation models for time-dependent turbulent cavitating flows. Acta Mech. Sin. 27, (2011)