Computational mesh consists of 110000 elements. At the nozzle inlet the mass flow rate , water volume fraction equals 1 and total gas temperature were set. This flow rate provides total pressure . The relative pressure at the computational domain outlet was set zero. At the initial stage the numerical calculation was performed using homogeneous approach with decreased by 10 times mass flow rate. Further calculations ware carried out using inhomogeneous approach. The implicit time step was . A greater time step leads to instability. The flow under consideration is unsteady. But measured parameters are averaged during rather long time period. So for correct comparison of experimental and numerical data the large number of time steps should be calculated. This number depends on the water velocity and time step. For present problem the required number of time steps is 40000. The calculation was progressed during 3 weeks. Figure 15 shows the distribution of gas Mach number near nozzle. It should note considerable heterogeneous velocity filed that is character for such a problem. There is small region near nozzle outlet where the flow velocity is greater by 1000 times than averaged velocity through the computational domain. It forces to set rather small time step. Figure 16 shows the distribution of water volume fraction at different time steps. For comparison there are photos of jets maiden in experiments in the figure 17. Fig.15 Gas Mach number distribution Fig.16 Water volume fraction distributions at different time steps Fig.17 Photos of gas jets inside cocurrent water flow, exposure shutter is 0.001sec These photos allow to compare the flow structure. Figure 18 shows experimental and numerically obtained gas jet shape as function of jet diameter and distance along jet axis. Fig.18 Dependence of jet diameter upon distance along jet axis Fig.19 Water volume fraction distribution along jet axis Figure 19 shows the comparison of water volume fraction distribution measured in the experiments and obtained in numerical calculations. There is good correlation between numerical and experimental data. The most important shortage of the method is the need of long time for calculations. It is required around 3 week machine time for axysimmetric problem solving. 2. SUPERCAVITY FLOW MODELING BASED ON SOLVING OF THE LAPLACE PROBLEM FOR THE EXTERNAL FLOW AND REYNOLD'S EQUATIONS FOR THE INTERNAL FLOW (GAS FLOW INSIDE THE CAVITY) In [7], it was developed a method, which allows for computations of the non-axsymmetric cavity shape including the gas jet flow inside the cavity and its interaction with the boundary. Unlike the method considered earlier, only the gas flow inside the cavity is numerically modeled. The external water flow is determined analytically using the slender body theory. Main theses of the work are outlined hereafter [7]. We will proceed from the following assumptions: - Gas and liquid are not get mixed, gases do not dissolve in water - The liquid evaporation can be neglected, because the saturated gas pressure is much smaller than the cavity pressure - The liquid velocity does not change on the cavity surface due to viscous interaction with the gas jet - Water splashes and their influence on the gas flow are neglect With these assumptions the gas flow in the cavity is similar to the flow in the tube. On the surface, the normal and tangent gas velocities equal to the normal and tangent liquid velocities, respectively.
The tube wall temperature equals to the water temperature, the heat flux has such a value that the gas flow in bottom part is isothermal. Hence the supercavity liquid flow problem at the presence of gas jets can be reduced to internal gas flow modeling in the tube. The cavity shape is determined by the flow field, more exactly by the static pressure distribution on the cavity boundary, and can be computed using analytical expressions which have good correlation with experiments. In the present work, the cavity shape is computed using the system of equations derived in [8]. The cavity shape is expressed as , where is relatively small non-axisymmetric cavity-surface deformation, is non-disturbed cavity radius, is shown in Fig. 20. Fig.20: Cavity and coordinate system Expanding the deformation into Fourier series we obtain the system of linear equations for the coefficients [8]: (9) Here , , are Fourier coefficients of small deformations, extra cavity numbers and, mass forces potential, respectively; the coefficient . If the disturbances are caused by a -periodic forcing, which is induced, for example, by equidistant nozzles, then we can consider only part of the cavity: (9) (10) , Using the system of equations (8) it is possible to compute the shape of the cavity deformed by different disturbing factors such as irregular static pressure distribution along the cavity boundary , external disturbing pressure field, gravity field, cavitator form difference from the circle etc. The undisturbed cavity shape can be calculated using the simple analytical expression, which has been verified by many experiments [4], [9]: (11) where is undisturbed cavitation number, and . The gas flow inside cavity can be obtained by numerical integration of the Reynolds system of equation and SST turbulence model. However the tube shape and gas flow are interdependent so the problem should be solved iteratively. At first the computation is performed for a certain initial tube shape. Then the new tube (cavity) shape is refined using the expression (8). The flow domain and computational grid are deformed in accord with this shape and computations are performed for the new geometry. This iteration process is continued until the cavity shape sets in. The final pressure distribution and cavity shape should satisfy the dynamics and kinematics conditions. Since this may require a large number of iterations, the process should be automatic. The ANSYS CFX 10.0 solver allows for deforming computational domain boundaries. The deformation law is set with the help of a user subroutine. The required values (pressure in mesh nodes here) are exported into the subroutine. The pressure data are used to calculate the new cavity shape and new mash coordinates via Eq. (8) and this information is passed back to the solver. A typical computational domain is shown in Fig.21. Since the problem is symmetric, only a half of region is used. Fig.21: Computational domain of gas flow inside cavity The nozzles are located at a certain distance from the cavitator. The nozzle inlet is the inflow boundary of computational domain. The boundary conditions are: - No-slip boundary condition on the surfaces of nozzle, body and cavitator - Mass flow rate and stagnation temperature at the nozzle inlet -
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