Simulation of Internal Flow Field in Pulse Thruster Unit

Pulse thrusters are widely used in the attitude control and flight trajectory control of rotating ammunition and vertical launching ammunition, which use own pulse impulse to change the attitude and trajectory of the missile. This paper simulates the internal flow field of a double-layer pulse thruster unit (PTU). The internal flow field of the PTU is simulated at various temperatures based on the three-dimensional compressible N-S governing equations and k-ω SST turbulence model. In addition, the influence of nozzle throat length and divergence half-angle on the performance of pulse T-thruster is analyzed. The simulation results demonstrate that the thrust output error of the PTU is minimum, when the nozzle throat length of the internal pulse thruster is 38 mm. Moreover, the stability of the PTU is improved.


Introduction
Pulse thrusters are widely used in the attitude and flight trajectory control of rotating ammunition and vertical launching ammunition [1,2].The pulse thrusters use their pulse impulse to change the attitude and the flight trajectory, which are used to improve the performance of projectile hit accuracy.According to the installation position of the pulse thruster, it is divided into the flight trajectory control pulse thruster and the attitude control pulse thruster.The trajectory control pulse thruster is usually installed at the centroid position of the missile, and the direct force acts on the missile centroid [3].However, the attitude control pulse thruster is usually installed on the head of the missile, and the direct force generates a moment relative to the center of missile centroid, so that the projectile attitude is adjusted [4,5].In addition, the pulse thrusters are also used to adjust the flight trajectory of vertical launch ammunition [6,7], the thrust generated by the pulse thruster is perpendicular to the projectile axis, and the combined thrust generated by two or more pulse thrusters can cause the missile to produce radial motion, which can quickly turn the missile into the predetermined trajectory [8].
Based on the above problems, this paper presents a PTU with double-layer structure, which is composed of certain number pulse T-thrusters.This Paper simulates the internal flow field of the internal and external T-thrusters at three working temperatures, when the nozzle throat lengths of the internal pulse T-thruster are different.The purpose is to ensure same thrust between the internal and external pulse thrusters.According to the integrated double-layer layout scheme described in Figure 1, we can note that the nozzle throat lengths of IPT and EPT are different.Figure 2 shows the schematic diagram of the EPT, and Figure 3 shows the schematic diagram of the IPT.The datas of interior ballistic calculation are pressure, thrust, working time, total impulse, etc.The temperatures of 20 ℃, 65 ℃and -40 ℃ are respectively selected to simulate the three possible working environments of the pulse T-thruster.Among them, 20 ℃ represents the room temperature working condition, 65 ℃ represents the high temperature working condition, and -40 ℃ represents the low temperature working condition.Table 1 shows the interior ballistic calculation results of the pulse T-thruster.F is equilibrium thrust, I is total impulse, and s I is specific impulse.

Model of simulation
As we know that the solid phase content is very small in combustion products of GATo-3 propellant [9] , combustion products can be regarded as pure gas phase mixture.Gas flow is the flow where the transportation phenomena of friction, thermal conduction and mass diffusion are included, and the process also includes a variety of complex physical and chemical phenomena.Ignoring the thermal transfer in the flow process, the whole working process is in the adiabatic mode.The flow is assumed as compressible, turbulent, and continuous.In this paper, the numerical simulation is performed based on the governing equations and turbulence model.

Governing equations
Navier-Stokes equations are the governing equation describing the continuous medium, which include the equations of mass conservation, momentum conservation and energy conservation.The three-dimensional compressible Navier-Stokes equations are given as follows [10]: The continuity equation is where  is the density, V is the vector velocity field, u , v, and w are the x, y, and z components of velocity,  is the vector operator.Momentum equation is The energy equation is where p is the pressure, T is the temperature, f is the body force.The supplementary equation is where  is the molecular kinetic viscosity,  is the second viscosity coefficient, and τ is stress tensor.
For a perfect gas, the equation of state is (5) where R is the specific gas constant.
The Isentropic equation is where c is the specific heat ratio.

Turbulence model
The k-ω Shear-Stress Transport (SST) turbulence model has been proved to be accurate and reliable for a wide class of flows, particularly for internal nozzle flows [11][12][13].The SST turbulence Model combines k-ε model and k-ω model through blending function 1 F .The Model is described as follows [14,15]:   10 2 1 max 2 , 10 where k is turbulent kinetic energy, ω is the turbulence frequency,  is the density and j u is the flow velocity, t  is turbulent eddy viscosity, y is the distance to the nearest wall, S is the invariant measure of the strain rate, 1 F and 2 F are blending functions.The constants are:

Mesh and boundary conditions
The quality of the mesh plays an important role in numerical simulation of finite volume method, which directly affected the accuracy of subsequent numerical analysis [16, 17] .The POINTWISE mesh generation software is used to divide the hexahedral element into structured meshes, because the hexahedral grid can provide higher accuracy and reduce the cost of finite element analysis and computational fluid dynamics [18, 19] .In addition, because the model is symmetrical, only half of the model is meshed, which will not affect the calculation results while reducing the calculation time.Figure 4 is the mesh distributions of IPT, and Figure 4 sets the boundary.Figure 5 is the mesh distributions of EPT.The boundary of EPT is the same as IPT.The 3D axisymmetric grids of nozzle 6 domain are generated used in the POINTWISE (version 18.3R1) [20], and the boundaries of the volume are also created used in the POINTWISE.According to the flow characteristics of the supersonic compressible fluid, the boundary conditions are imposed as follows: pressure boundary conditions for the inlet and outlet of the nozzle, respectively, and the no-slip and adiabatic boundary conditions are specified for the walls.
Pressure inlet: the total pressure and total temperature are set to be 40.58MPa and 3394 K in inlet 1 and inlet 2 at 20 ℃.The equilibrium pressure is 40.58MPa at 20 ℃.The total pressure and total temperature are set to be 49.51MPa and 3394 K in inlet 1 and inlet 2 at 65 ℃.The equilibrium pressure is 49.51 MPa at 65 ℃.The total pressure and total temperature are set to be 29.87MPa and 3394 K in inlet 1 and inlet 2 at -40 ℃.The equilibrium pressure is 29.87 MPa at -40 ℃.The data of equilibrium pressure is shown in the Table 1.The combustion temperature of GATo-3 propellant is 3394 K.
Pressure outlet: because the outlet velocity of the nozzle is supersonic, all parameters are extrapolated.The outlet pressure is 1 atm and the temperature is 300 K in the initial calculation.
ANSYS FLUENT (version 19.2) [21] is selected as the computational fluid dynamics tool to simulate the internal flow field of the pulse T-thruster in this study.

Results and Discussion
The thrust output of IPT is adjusted by changing the nozzle throat length and the divergence half-angle, to make IPT and EPT have the same performance.Due to the limitation of IPT installation, the nozzle throat length varies from 1 mm to 45 mm, and the divergence half-angle varies from 12°to 39°.The internal flow field of thruster is simulated and analyzed under three temperature conditions in this paper.The simulation results are shown in Figure 6 and Figure 7. Figure 6 shows the velocity contours and streamlines of the IPT symmetry plane when the nozzle throat length is varied from 1 mm to 45 mm under room temperature.Figure 7 shows the mean pressure contours of outlet, when the nozzle throat length is varied from 1 mm to 45 mm under room temperature.As shown in Figure 6, streamlines are basically symmetrical and smooth, there is no vortex in the flow field of the thruster under different nozzle throat lengths, and the velocity of outlet is affected by the nozzle throat length.As shown in Figure 6 (b), when the nozzle throat length is 20 mm, the maximum velocity is 2286 m/s.Correspondingly, the mean velocity of outlet reaches its maximum 2207.52 m/s, which is obtained by Area -Weighted Average in Fluent.In contrast, Figure 6 (d) shows that when the length of the nozzle is 45 mm, the maximum velocity is 2140 m/s.Correspondingly, the mean velocity of outlet reaches its minimum 2041.58m/s.The range of mean velocity of outlet is 2041.58-2207.52m/s.Similarly, high temperatures and low temperature are simulated and analyzed, respectively.The simulation results are shown in Table 2.The parameters include mass flow rate, mean velocity and mean pressure of nozzle outlet.
According to the thruster theory, the thrust output of the thruster can be calculated from the following formula [22]: A is the cross sectional area of the nozzle outlet, a p is the atmospheric pressure.
Therefore, according to the mass flow rate, mean velocity and mean pressure at outlet calculated by the simulation shown in Table 2, the thrust of EPT can be calculated by formula (15), and the calculation results are shown in Table 3.In the similar process, the EPT is simulated and analyzed, and results are shown in Table 4. Compared with the data in Table 2 and Table 4, it is clear that the nozzle throat length directly affected the performance parameters of IPT.As discussed earlier, the thrust output of IPT can be changed by adjusting the nozzle throat length and divergence half-angle.Therefore, what we are most concerned about here is how large the nozzle throat length and divergence half-angle of IPT are selected to minimize the thrust output error of IPT with EPT.The comparison of the thrust output between IPT and EPT is shown in Figure 8.

2 .
Structure and Interior ballistic datas of the PTU Figure 1 shows schematic diagram of double-layer structure the PTU, which consists of a finite number of pulse T-thrusters spaced equally around the circumference of the rocket missile.In Figure 1, the blue circles represent the external pulse T-thruster (EPT), and the red circles represent the internal pulse T-thruster (IPT).The axis of combustion chamber is perpendicular to the nozzle axis for each pulse T-thruster.

Figure 1 .
Figure 1.Schematic diagram of double-layer structure the PTU.

Figure 2 .
Figure 2. Schematic diagram of the EPT.Figure 3. Schematic diagram of the IPT.

Figure 3 .
Figure 2. Schematic diagram of the EPT.Figure 3. Schematic diagram of the IPT.

Figure 7 .
Figure 7. Pressure contours of IPT.Figures 7 (a)-(d) show the outlet pressure contours with nozzle lengths ranging from 5 mm to 45 mm.From the simulation results, the pressure contours of nozzle outlet are different with different nozzle throat length.The mean pressure of outlet varies from 1.582 MPa to 2.075 MPa, which is obtained by Area -Weighted Average in Fluent.Similarly, high temperatures and low temperature are simulated and analyzed, respectively.The simulation results are shown in Table2.The parameters include mass flow rate, mean velocity and mean pressure of nozzle outlet.According to the thruster theory, the thrust output of the thruster can be calculated from the following formula[22]:

Figure 8 .
Figure 8.Comparison chart of the thrust.5.ConclusionsThe PTU consists of double-layer pulse T-thrusters with different nozzle throat lengths.Based on the three-dimensional compressible N-S governing equations and the k-ω SST turbulence model, the simulation of internal flow field of pulse T-thruster with different nozzle throat length is carried out under three different temperature conditions.The simulation results show that thrust output of the pulse T-thruster decreases with the increase of the nozzle throat length under different temperature conditions, which is about 4.4%.When the nozzle throat length of the IPT is 38 mm, the divergence half-angle is 30 degrees, and the thrust output of the IPT and EPT is equivalent, the thrust output of the double-layer structure the PTU is stable.FundingProject supported by the Natural Science Foundation of Guangdong Province, China (Grant No. 2021A1515011401).Project supported by the Key Research Program of the Higher Education Institutions of Department of Education of Guangdong Province, China (Grant No. 2021ZDZX1012).Project supported by the Research Program of the Higher Education Institutions of Department of Education of Guangdong Province, China (Grant No. 2019KTSCX180).

Table 1 .
Datas of Interior Ballistic Calculation.

Table 2 .
Calculation results of IPT.

Table 4 .
Calculation results of the EPT.