In order to eliminate the influence of gravity on the jet flow field, a vertically placed experimental device was designed. The gas was jetted vertically upwards and the structure of the experimental device is shown in Fig. 1, mainly combined by the base, the combustion chamber, the connecting pipe, the nozzle, the tail pipe, the liquid–water column, etc. For ease of installation, the liquid–water was enclosed in a thin-walled cylindrical plastic tube and placed in the tail pipe. The working principle of the experimental device was using the electric ignition device to ignite the propellant filled in the combustion chamber. High temperature and high pressure gas was generated when the propellant burned rapidly. When the pressure in the combustion chamber increased to a certain level, the diaphragm at the throat of the nozzle was flushed to broken, and the gas was accelerated by the nozzle to form a supersonic gaseous jet. If the liquid–water column was placed in the tail pipe, the gaseous jet would impinge the liquid–water column and sprayed into the atmosphere in the process of interacting with water. The experiment was carried out under two conditions: (i) the investigation of gaseous jet flow field without liquid–water column; (ii) the investigation of gas–liquid mixed flow field formed by gaseous jet impinged the liquid–water column.

Fig. 1.

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	Fig. 1. The schematic diagram of experimental apparatus.

In order to obtain the working characteristics of the rocket engine and flow field characteristics under two experimental conditions, the relevant measurement system was designed in this experiment. At point 1 and point 2 shown in Fig. 1, a KISTLER-601 H piezoelectric pressure sensor manufactured by KISTLER Company was placed respectively. The DEWE-4010 transient data recorder made by the company of LABG was used to record the change process of pressure in the combustion chamber and tail pipe. Due to the complexity of the unsteady gaseous jet flow field and the gas–liquid mixed flow field, each point in the flow field had a strong random fluidity, and it was difficult to mark and track the flow at a certain point. Therefore, this experiment used the image acquisition system to record the flow field formed under each experimental condition, and qualitatively obtained the change of the flow field structure. The system consisted of a high-speed camera and an image processing computer. In order to reduce the error caused by the external environment to the experimental results, this experiment was carried out in an open field, the ambient temperature was about 25 °C, the air humidity was less than 50%, and the wind speed was less than level 3. The experimental device was installed on a concrete column platform at a certain height from the ground.

Figure 2 shows the expansion process of gaseous jet and gas–liquid mixed jet in the atmospheric environment captured by high-speed camera. It can be seen from the figure that in the case of a liquid-free water column, the gaseous jet expanded in the axial direction significantly faster than the radial expansion speed. Due to the higher temperature of the gas, the interior of the jet flow field exhibited high brightness characteristics. After t = 35 ms, A large amount of white smoke began to appear at the exit of the tail pipe and at the edge of the jet flow field, which was formed by the condensed particles in the propellant combustion products.

Fig. 2.

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	Fig. 2. The distribution of jet flow field in the atmospheric environment at different times. (a) The expansion process of gaseous jet in the atmospheric environment. (b) The expansion process of gas–liquid mixed jet in the atmospheric environment.

Under the experimental condition that the liquid–water column was placed in the tail pipe, the gas would first impinge the liquid–water column when the gas flowed out from the nozzle. After that, the gas and the liquid–water jointly ejected from the tail pipe in the process of strong mixing and formed the gas–liquid mixed jet flow field. Due to the hindrance of the water column, the time of gas–liquid mixed jet ejecting from the tail pipe was lagged behind that of gaseous jet. The axial direction of the jet flow development speed also became slow. As shown in Fig. 2(b), at t = 36 ms, the outer contour of the gas–liquid mixed jet flow field gradually began to appear jagged, showing strong instability at the gas–liquid interface. This instability would cause the liquid to break up, and a large number of small droplets appeared and increased the vaporization surface of the liquid, which helped the liquid–water vaporize to absorb more gas energy, further reducing the temperature of the flow field. The high brightness characteristics were no longer visible compared with the gaseous jet flow field without the liquid column. At the same time, due to the atomization of the liquid–water, the entire flow field area was covered by a large amount of water mist, so that the scattering and absorption of light by the particles was not enough to be detected. The white smoke that appeared in the experiment state without liquid–water column no longer appeared.

Figure 3 shows the comparison curves of pressure changed in the combustion chamber and tail pipe under the experimental conditions with or without liquid–water column. Under the experimental condition containing liquid–water column, the pressure in the combustion chamber increased overall, but two pressure oscillations with large amplitude occurred when t = 34.75 ms and t = 35.27 ms. The instantaneous pressure increased by 17.73% and 17.93% respectively and dropped rapidly after reaching the peak. At the same time point, the pressure in the tail pipe also had two pressure oscillations with large amplitude, but after the second oscillation, the pressure in the tail pipe decreased gradually. Under the experimental condition without liquid–water column, the changes of pressure in the combustion chamber and tail pipe were relatively stable, and there were relatively gentle ascending, working, and descending stages. However, the pressure rising stage in the combustion chamber between t = 41 ms and t = 42 ms may be caused by the uneven combustion of propellant at the end of rocket working.

Fig. 3.

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	Fig. 3. Pressure change curves in combustion chamber and tail pipe measured by experiment. (a) Pressure change curves in combustion chamber. (b) Pressure change curves in tail pipe.

The process of gaseous jet impinging on the liquid–water column involves complex multiphase flow problems. In order to achieve effective calculation, this article adopts a simplified physical model and proposes the following assumptions: (I) the actual three-dimensional flow problem is simplified into a two-dimensional turbulent flow, which is considered as an unsteady expansion process; (II) the combustion product of propellant is assumed as an ideal gas with a single component, and a frozen isentropic flow model is adopted, that is, the composition of the combustion product of gas phase does not change with the change of temperature and pressure; (III) only the gas phase is the compressible phase, and the liquid phase and the water–vapor phase are both incompressible phases; (IV) the heat exchange between the gas and the walls is not considered because of the gas in the experimental device flowed at a high speed. In other words, the walls are regarded as adiabatic and non-slip walls.

According to the assumptions of the above physical models, the following numerical equations are established.

The continuity equation is defined as 1 In Eq. (1), t is the time, is the density of the mixture, is the Hamiltonian operator, is the average velocity of the mixture, and m_S is the custom quality source item. According to the combustion equation of propellant used in the experiment, m_S is expressed as 2 where is the density of propellant, a is the burning rate coefficient of propellant, p_v is the average pressure in the combustion chamber, b is the burning rate index, and S is the burning surface area of propellant.

The momentum conservation equation can be obtained by summing the momentum equations of each phase, which can be expressed as 3 In Eq. (3), p is the local pressure at each node, is the mixed viscosity, is the gravity acceleration, is the volume force, k stands for the k phase, and n stands for the total number of phases. There are three phases in the numerical models of this article, namely, the gas phase at k = 1, the liquid phase at k = 2, and the water–vapor phase at represents the volume fraction of the k phase, is the density of the k phase, and is the drift speed of the k phase.

The energy conservation equation is defined as 4

For compressible phase, – . For incompressible phase, , where h_k is the sensible enthalpy of the k phase. k_eff is the effective heat conductivity defined according to the turbulence model. is the thermodynamic temperature of the mixture. The first term on the right side of Eq. (4) represents the energy transfer due to heat conduction, and S_E represents the energy change caused by gas mass addition and physical phase change of water.

Since only the gas phase is a compressible phase, the state equation of each phase is as follows: 5

In Eq. (5), R stands for molar gas constant and T₁ stands for instantaneous temperature of the gas phase.

In order to simulate the vaporization process of water and the condensation process of water–vapor, the physical phase change equations of water are used to calculate it. The state of water at each compute node is determined by the pressure and temperature at that node. When the temperature is higher than the saturation temperature of the liquid–water at the local pressure, the liquid–water absorbs energy and is vaporized into water–vapor. When the temperature is lower than the saturation temperature of the liquid–water at the local pressure, the water–vapor releases energy and condenses into liquid–water. The physical phase change equations of water are as follows.

The liquid–water–vaporization equation is defined as 6 The water–vapor condensation equation is defined as 7 In Eqs. (6) and (7), m₂ and m₃ stand for the vaporization rate of liquid–water and the condensation rate of water–vapor, respectively, T_sat is the saturation temperature of liquid–water at the local pressure, T₂ and T₃ are the instantaneous temperatures of the liquid phase and the vapor phase, respectively, and β is the vaporization coefficient. According to the above formulas, the instantaneous net vaporization rate of liquid–water can be calculated as 8 The energy change caused by the phase change is 9

In Eq. (9), S_V is the energy change caused by phase change, and is the latent heat of vaporization of saturated water. The energy change caused by mass addition is expressed as 10 11 In Eqs. (10) and (11), S_m is the change in energy due to mass addition, C_p is the specific heat of constant pressure, and T_p is the burning temperature of constant pressure. In numerical calculation, the energy change S_E is substituted into the energy equation by the mode that adds source terms.

In the calculation of multiphase flow, in addition to the continuity equation of the mixture, the volume fraction of each phase should be calculated by the auxiliary continuity equation of gas phase and water–vapor phase.

The gas phase continuity equation is defined as 12 The water–vapor phase continuity equation is defined as 13 Due to the sum of the volume fraction of all the phases is 1, the volume fraction and of the gas phase and the water–vapor phase can be obtained by solving the above continuity equations, thereby obtaining the volume fraction of the liquid phase as .

Since the formed flow field was a three-dimensional axisymmetric structure, this article has simplified the three-dimensional flow field into a two-dimensional computational domain, including the combustion chamber, nozzle, tail pipe, and external standard atmospheric space, as shown in Fig. 4. The geometries of the combustion chamber, nozzle, and tailpipe were consistent with the dimensions in the experimental structure. The size of the external atmospheric space was (length × width), where d was the diameter of nozzle throat. In order to observe the change rule of physical quantity in combustion chamber, one monitoring point was set on the wall of combustion chamber. The location of the monitoring point is the same as that of test point 1 in experimental study. When using non-reflective boundary conditions, this computational domain can ensure that the real flow field parameters can be obtained without the computational boundary interference.

Fig. 4.

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	Fig. 4. The computational domain.

In the experiment of gaseous jet impinging on the liquid–water column, a pressure limiting diaphragm was placed at the throat of the nozzle. When the pressure in the combustion chamber reached the membrane breaking pressure, the gas flowed out from the nozzle and impinged the liquid–water column. Therefore, in this article, the whole calculation area was divided into two parts, and the combustion chamber and the nozzle contraction section were divided into region 1. The region 1 was set as a region continuously producing gas by using the method of defining source items to simulate the combustion process of the propellant in the combustion chamber. The nozzle expansion section, the tail pipe, and the external atmospheric environment were divided into region 2. At the time of initialization, the corresponding area in the tail pipe was designated as the water area according to the size of the liquid–water column in the experiment, and the other regions were the standard static atmospheric environment. The boundaries of the combustion chamber, the nozzle, and the tail pipe were set to the wall boundary condition, and the boundaries of the external atmospheric environment were set as the pressure outlet boundary condition. In this article, the finite volume method was used to discretize the governing equation. The turbulence model used a model with swirl correction (realizable model) to describe the process of interaction between gaseous jet and liquid–water. The wall surfaces were selected without slip and adiabatic wall conditions, and the near wall adopted the standard wall function method to treat turbulence. The pressure-implicit with splitting of operators (PISO) algorithm was used to solve the coupling of pressure and velocity, volume fraction equations were discrete by using the QUICK format, and the pressure term was discrete by using the PRESTO method. The other diffusion terms, convection terms, and the discrete formats of turbulent k and ε equation all adopted the first-order upwind style.

Based on the computational domain established in the previous section and the numerical calculation method, the unsteady process of gaseous jet impinging on the liquid–water column was numerically calculated in this paper. Due to the high sampling frequency and large amount of data stored in the data acquisition system, this paper selected the pressure data of one hundred time points in the experimental results for comparative analysis with the numerical calculation results. Figure 5 shows the comparison between the experimental and the numerical pressure results at the monitoring point. It can be seen from Fig. 5 that the numerical calculation results are consistent with the change trend of the experimental results relative to time, which can reflect the pressure change law at the monitoring point in the process of gaseous jet impinging on the liquid–water column, indicating that the mathematical model and calculation method established in this paper were reasonable and feasible. As can be seen from Fig. 5, when the pressure in the combustion chamber reached the stable working period, the pressure values obtained by numerical calculation were slightly higher than that measured in the experiment. This may be due to the incomplete burning of propellant in the combustion chamber during the experiment study and the mixed jet inevitably caused some energy losses in the actual flow process. There are twice obvious pressure oscillations with large amplitude in both experimental and numerical results. The causes for pressure oscillations at the monitoring point would be analyzed according to the following numerical results.

Fig. 5.

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	Fig. 5. The comparison diagram of experimental results and numerical results.

Since the liquid–water column had little influence on the working performance of rocket after being completely pushed out of the tail pipe by gaseous jet, this paper only selected the calculation results within 2.5 ms from the beginning of the impingement for analysis and research. Figure 6 shows the distribution of water volume fraction in the gas–liquid mixed jet flow field at different times.

Fig. 6.

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	Fig. 6. The distribution diagram of water volume fraction at different times.

Figure 7 shows the time-dependent curve of the development distance of the cavity along the axis in the liquid–water column. Combining Figs. 6 and 7, it can be seen that when t = 0.2 ms, the gaseous jet began to impinge the liquid–water column. The gas–liquid interface produced obvious displacement, and the gas gradually formed a cavity in the liquid–water column. Due to the small amount of gas generated at this time, the pressure in front of gas–liquid interface was low, resulting in a slow increase in the speed of the cavity. However, with the development of the flow field, the development speed of the cavity increased significantly at t = 0.5 ms and t = 1.2 ms, which was consistent with the time when the two pressure oscillations in the combustion chamber reached their peak values. When t = 1.2 ms, the head of the cavity presented a regular semicircle. After that, the water moved forward along both sides of the pipe wall under the push of the gaseous jet, and the shape of the gas–liquid interface did not change significantly during the movement. When t= 2.5 ms, only a little residual water in the tail pipe adhered to the pipe wall, and the flow field started to develop along the radial direction without the constraint of the pipe wall. Because the gaseous jet pushed the liquid–water column to move in the tail pipe for only 2.5 ms, the action time was relatively short, so the liquid was not vaporized completely and most of the water was still sprayed into the external atmosphere environment.

Fig. 7.

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	Fig. 7. The cavity displacement curve.

The cloud diagrams of pressure distribution in the computational domain at different times were shown in Fig. 8. The moment when the gaseous jet broke through the diaphragm at the nozzle throat was taken as the initial time of numerical calculation. At this time, the membrane breaking pressure of 5 MPa was evenly distributed in the combustion chamber.

Fig. 8.

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	Fig. 8. The cloud diagrams of pressure distribution at different times.

As can be seen from the figure, when the diaphragm was broken, the gaseous jet flowed rapidly into the expansion section of the nozzle, resulting in a pressure decrease near the nozzle inlet in the combustion chamber. At this time, the gas near the bottom of the combustion chamber flowed relatively slow, and the pressure near the bottom of the combustion chamber rose rapidly with the rapid combustion of the propellant. Due to the inertia effect of the high-density water, the gaseous jet was blocked by liquid–water column, and a large amount of gas accumulated in the nozzle expansion section, which leads to the pressure rise in this region and the shock wave formation in the nozzle expansion section. With the combustion of propellant and the flow of gas, the pressure after shock wave increased continuously, and the phenomenon of pressure wave back propagation was formed. Under the action of reverse pressure wave, the shock wave moved along the nozzle expansion section towards the nozzle throat. Therefore, within the time frame from t = 0.3 ms to t = 0.5 ms, the pressure near the nozzle inlet in the combustion chamber increased sharply and continuously extended to the combustion chamber interior. When t = 0.5 ms, the pressure reached the maximum value, which also corresponded to the wave peak of the pressure oscillation of the first time at monitoring point 1. When the pressure at the nozzle inlet and the pressure at the nozzle outlet reached a certain ratio, the gas could flow smoothly from the combustion chamber into the nozzle again, so the pressure in the combustion chamber decreased quickly. However, since the liquid–water only produced very small motion at this time, the gas once again accumulated in large quantity between the nozzle throat and the gas–liquid interface. The reverse pressure wave caused the shock wave to move in the opposite direction again and the flow of gas was blocked again, resulting in the emergence of the second time pressure oscillation at t = 1.2 ms. At this time, the peak value of the second time pressure oscillation also increased because the gas production of propellant combustion was significantly increased compared with the first time pressure oscillation. Thereafter, the liquid–water column moved rapidly to the external atmospheric environment under the push of high-pressure gas. With the continuous increase of the free volume in the tail pipe, the pressure in the combustion chamber gradually reached a relatively stable working state. During this period, the shock wave in the nozzle expansion section also began to move from the nozzle throat to the nozzle outlet. Until t = 2.5 ms, when most liquid–water had been ejected from the tail pipe, the shock wave had moved from the nozzle expansion section to the tail pipe interior. After that, the influence of the interaction between gas and liquid–water on the working performance of the rocket was trivial.