The simulations are performed with various non-dimensional volume flowrates . Non-dimensional jet impulse was less than . Each flow case was computed as homogenous and inhomogeneous. The boundary conditions are: ¬ Two inflows. Water velocity and water volume fraction equal 1 on first inflow (Fig. 1). Gas mass flowrate and gas volume fraction equal 1 on second inflow (Fig. 4) ¬ No slip adiabatic condition on cavitator surface ¬ No leakage condition on upper side of the domain ¬ Hydrostatic pressure on the outflow boundary Computations of unsteady flow are performed with the time step . The maximum Courant number is less than 40, its average value is about 1. Computed flows are essentially unsteady. Figure 5 show water volume fraction distributions at some time steps computed using the inhomogeneous method . The blue color corresponds to 100% water. The re-entrained jet is at the cavity bottom. The jet flows inside cavity toward gas jet, breaks and forms splashes that are passed away from the cavity. Cavity begins rising without gas loss but at one moment gas portions breaks away and cavity size decreasing. Gas loss is unsteady. Fig. 5: Water volume fraction distributions at some time steps, . Inhomogeneous flow Figure 6 shows cavitation number along cavity computed using the inhomogeneous method. The cavitation number little varies along the cavity. Fig. 6: Cavitation number distribution along cavity, inhomogeneous flow, Figure 7 shows time dependence of the cavitation number averaged along cavity. Fig. 7: Time dependence of the cavitation number averaged along cavity length, Figure 8a and 8b show distributions of inside and outside cavity computed using the homogenous and inhomogeneous methods. Water velocity is m/s, non-dimensional gas volume flowrate is . Fig. 8a: distribution, homogeneous flow, Fig. 8b: distribution, inhomogeneous flow, Figure 9a and 9b show the water volume fraction distributions. The blue color corresponds to 100% water. Fig. 9a: Water volume fraction distribution, homogeneous flow, Fig. 9b: Water volume fraction distribution, homogeneous flow, A lot of experiments show that pressure is almost constant along the cavity and the cavity shape is almost symmetry relative middle section plane [4]. Such results are obtained using the inhomogeneous method. Non-uniform pressure distribution and non-symmetry cavity are obtained using the homogeneous method. So the important conclusion is followed. It is not valid modeling supercavitational flow using homogeneous approach. The results of computations are summarized in Table 2 containing theoretical and numerical values of the drag coefficient, the middle section radius, , and cavity half-length, . Theoretical data are calculated using value from numerical simulation (1), (2). Numerical and are time averaged. Table 2 Numerical Approach Numerical Theoretical 2 Homog. 0.048 4.2 28 4.1 38 2 Inhomog. 0.041 4.3 39 4.3 45 3 Homog. 0.039 4.7 35 4.7 47 3 Inhomog. 0.031 4.8 45 4.9 61 4 Homog. 0.035 5.1 40 5.0 53 4 Inhomog. 0.026 5.2 59 5.2 71 5 Homog. 0.031 5.3 45 5.3 61 5 Inhomog. 0.023 5.6 62 5.5 80 The is modeled rather well in both homogenous and inhomogeneous approach as contrasted to that is quite different from the theoretical data. Figures 10a and 10b show gas temperature distribution in case when the temperature of the inflowing gas is and correspondently. Water velocity is m/s, , water temperature is K. Fig. 10a: Temperature distribution. K Fig. 10b: Temperature distribution.
K Gas intensively cools up to the water temperature and its flow could be considered as isothermal along most cavity part. Logvinovich [4] analyzing experiments concluded that the cavity bow profile behind flat disk almost does not depend on pressure when and could be expressed. , (8) where is distance from cavitator along cavity axis, is cavity radius. Above cavity sizes obtained theoretically and numerically are compared. Figure 11 shows compare of the cavity bow profiles obtained numerically and using (8). There is good correspondence between numerical and theoretical data. Fig. 11: Profiles of the cavity bow The fact that cavity bow profile and radius of its middle section are in good correspondence with theoretical and empirical data allows us stating that this method provides rather correctly model supercavitational flow. Although the essential difference between lengths computed numericaly and theoretically is there. T shows that further research should be performed. The major shortcomings of this method are high computational resources and slow convergence of numerical solution during integration of hydro-gas-dynamics equations. The supercavitational flow modeling with gas jets inside cavity could be more convenient using another method that is considered in next part of this paper. 1.3. Cavity modeling at body water entry This method allows modeling the cavity geometry development and splashes formation while body enters the water. To model this problem the body was fixed in computational domain and water level grew. Figure 12 show computational domain for water entry problem modeling. Water inlet is at lower horizontal boundary. The water velocity is directed vertically. Fig.12 Computational domain for water entry modeling The entering of the wedges with angle 15° and 30° at bottom was calculated. The results are shown on Fig.13. The splashes shapes are rather similar to experimental investigations. Fig.13 Splashes development at body water entry Fig. 14 shows the compare of the force acted upon the wedge with angle 15° at the bottom while it enters the water with experimental data by V.P. Sokolyansky (TsAGI). Fig.14 Vertical force acted upon the wedge with angle 15° at water entry as function of nondimensional immersion At nondimensional immersion less then there is good agreement between numerical and experimental data. 1.4. Gas jets submerged in the cocurrent water flow This method also can be used for modeling the flow of the gas jets submerged in the cocurrent water flow. There are no enough works devoted to this problem but in some scope this problem is rather important. Experimental investigations of such a flow are devoted Y.F. Zhuravlev's work [10]. There are nozzle with gas jet pumped through it is immersed in water flow. Dynamic pressures, water volume ration along nozzle axis, water flow velocity, gas total pressure in jet were measured during experiments. Also flow was photographed. This allows to obtain jet contour. In present work the attempt to model such a flow numerically is done. The base data for modeling: - Incoming water velocity is ; - Depth is ; - Nozzle critical section diameter is ; - Nozzle outlet cross-section diameter is ; - Nozzle opening angle is ; - Total gas pressure before nozzle critical section is . The problem is considered as axisymmetric.
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