The article is devoted to the numerical modelling of the force and thermal effects acting on a submarine’s missile silo during an underwater missile launch. Experimental studies are not sufficient for determining all the parameters of interest and elucidating the pattern of emergency situations. The correctness of the developed numerical method for modelling the loads acting on a submarine during an underwater missile launch was confirmed by comparing the results of the conducted numerical calculations and obtained experimental data, which were found to be in good agreement. Two-phase flow patterns were obtained; force and temperature effects were determined during post-start silo flooding with water.

The problem related to determination of hydrodynamic loads acting on the missile and on the submarine during underwater launch remains urgent in development of modern sea-launched ballistic missiles. For now, theoretical solution to the problem requires the following subproblems to be solved: determination of loads acting on the missile during launch and determination of loads acting on the submarine only. Based on the determined loads, requirements to the submarine’s hull structural strength, control and stabilization systems can be established. This work describes an endeavour to carry out numerical simulation of power loads acting on the channel during continuous motion of the piston followed by flooding of the channel. This model allows to determine power and thermal loads acting on the channel body. Therefore, the model can be used to determine power and thermal loads acting on the submarine’s launching tube. The reliability and accuracy of the numerical simulation method under development can be confirmed by comparing calculations and experimental data [1].

The entire process of hydro- and gas dynamics effect on the submarine can be divided in three successive steps:

Figure 1 shows two computational models representing missile extraction out of the launching tube and entering the approach flow. The launching tube diameter is equal to the missile diameter as shown in Figure 1a. The model representing missile extraction out of the launching tube is similar to the experiment described herein. Figure 1b shows the model with an annular gap between the missile and the tube. The missile travels through the launching tube along the fixing-driving bands [2].

There are various models describing the effect of the initial gas temperature on the oscillating process characteristics (force action). In this work, we study the following steady-state model of the process. Let us assume that when water mist (water mass dmж) is mixed with gas, it flashes out and the generated vapour is heated up to the gas temperature [3].

Of course, in this case the heat exchange with the channel walls can be ignored too, similar to the process with “disabled” phase transformations.

The energy equation for the gas-vapour thermodynamic system can be represented as follows:

where Cv – specific heat at constant volume; mг – gas mass; mж – water mass; T – initial temperature in vessel; Tж – water temperature;

r'(Tж) – internal heat of phase transformation, indicating a change of internal energy of fluid during evaporation:

where r(Tж) – latent heat of phase change, indicating enthalpy change. Values of partial gas pressure Pг and partial vapour pressure Pn are calculated using the equation of ideal gas state:

The equation (1) can be used for the state until vapour remains unsaturated.

The experimental simulation of the process of filling the channel with fluid was carried out according to the diagram shown in Figure 2. The piston (2) is inserted in the channel (1) with the diameter D = 55 mm and height L = 450 mm. There is fluid (4) at hydrostatic pressure Pж above the piston, and the gas volume (3) with internal pressure equal to Pго =~Pвх below the piston. At time t = 0 the piston suddenly moves upward while the gas under the piston is expanded; once the piston leaves the channel, the channel starts to be filled with fluid. Pressure disturbance that occurs during filling is measured at the channel bottom by the LKh610 piezoelectric pressure sensor. The sensor’s self frequency is 15 kHz with the measurement error of 2 %.

To solve the hydro- and gas dynamics problem, the control volume method (CVM) [4][5][6][7] can be applied, based on conservation equations in the integral form:

where the left-hand side of the equation contains transient and convection terms, and the right-hand side contains diffusion and source terms. The equation also includes the following parameters: Ф – random variable, the value of which depends on the equation under consideration (for example, in the equation of motion, Ф =~ V, in the energy equation, Ф = cpT); Г – diffusion coefficient for value Ф; Q(p, Fm) – source term that may contain either the components of mass forces or the components of differential pressure.

Simulation was carried out in a transient 3D layout with regard to the multi-phase behaviour of the medium and the presence of gravity force, with implementation of the CVM based on the CFD package. The system of equations was closed using two-parameter turbulence model k-ε. The VOF (Volume Of Fluid) model was used for description of interphase interaction. We used a hexahedral and tetrahedral grid regenerated by means of the layer-by-layer build-up process.

The interphase interaction is considered by means of the Euler’s homogeneous model and the free surface model – this combination is used for flows with the free phase contact area. These models directly determine the solution region for the interphase boundary surface by placing a special boundary condition in it. A common equation setup is solved for both phases (water and gas), plus an individual computation of the boundary where these phases interact.

Using the developed methods, we carried out a numerical simulation of the process of piston extraction out of the channel for two conditions:

а) Pж > Pго;

b) Pж < Pго.

The channel dimensions are similar to those during experiment (D = 55 mm, L = 450 mm).

a) Figure 3 shows comparison of the calculation results and the experimental data on excessive gas pressure in the channel. Time count is initiated once the piston starts moving in the channel.

According to Figure 3, we may conclude that the process of filling the channel with water jets is accompanied by pressure oscillations. Oscillation damping occurs approximately at t = 350 ms. Then (at t > 350 ms) gas bubbles emerge in the channel.

Figure 4 shows the computed video record of the process of piston extraction out of the channel.

Figure 5 shows the video record of channel flooding with water.

Numerical simulation of processes allows to calculate the channel flooding process during gas heating. Figure 6 shows channel force action dependence diagrams when gas is heated and not heated.

When the channel is filled with hot gas, an increase in the contact area between gas and fluid leads to intense evaporation with a sharp drop of pressure and temperature. This eventually leads to higher peak pressure values and to a longer period of their oscillations.

b) The work is devoted to numerical simulation of the process of piston extraction out of the channel at Pж < Pго. This model drastically differs from the experiment.

Figure 7 shows the video record of the process of piston extraction out of the channel followed by channel water flooding at Pж < Pго.

First, a gas bubble with its shape similar to hemisphere is formed, then a cylindrical gas cavity is formed below the piston bottom. Due to gas rarefaction in the channel, the gas cavity boundaries close up. A bottom gas cavity is formed below the piston bottom while a fluid jet starts to flow out along the channel axis until it reaches the channel bottom, then it stops, and flooding of the channel with water jets begins. There are time-dependent pressure variations at the bottom while a bottom cavity is periodically separated from the piston aft end.

Figure 8 shows the dependence of excessive gas pressure in the channel on time. Time count is started once the piston comes out of the channel.

The following conclusions can be made based on completed numerical calculations and theoretical studies.

1. Good agreement between calculations and experiment results proves the reliability and accuracy of the developed method of numerical simulation of loads acting on the submarine’s launching tube during missile launch (Fig. 4).

2. For further numerical simulation of loads acting on the submarine’s launching tube during missile launch, it is necessary to pay particular attention to the launching tube architecture and to the flow field of the fluid passing around the tube body.