It was used for computations of the flow around a supercavitational missile. In this configuration there were gas blowing to the cavity on the missile bow and high-pressure gas jets from the engine into the cavity bottom. In [2] it was added an option to simulate the body dynamics simultaneously with modeling of the cavity flow. These papers open up opportunities for numerical simulations of cavitational flows. Nevertheless issues of correct modeling of the cavity shape and gas loss were not treated in detail. Moreover there was no discussion of how correctly the gliding forces acting on the missile were computed. An objective of this work was to develop a numerical method similar to that reported in [1] and check its accuracy in the case of supercavitational flow behind a flat disk. The main criterion is correct prediction of the cavity shape (the middle section radius and the cavity half length ). These sizes depend on the cavitational number , where is hydrostatic pressure, is pressure in the cavity, is water density, is undisturbed water velocity. At low the values of and can be calculated as [4]: (1) (2) where is the cavitator radius, is drag coefficient at , and the coefficient . The multiphase Reynolds system of equations with SST turbulence model was integrated numerically using the Ansys CFX 10.0 solver. The phase to phase interactions (forces acting from one phase to other, heat transfer, condensation and vaporization) were treated as source terms in the right-hand sides of hydrodynamics equations with aid of a user subroutine. In many papers the supercavitational flow is considered as homogenous. This approach is based on assumption that the phase velocities are equal. Consequently there is only one source term in the continuity equation. It is so called a cavitational model describing mass flow-rate at condensation and vaporization. This approach allows for decreasing of the number of equations. The problem formulation is well described in [1-3]. Hereafter is water volume fraction of -phase, , , , , , is its velocity, density, pressure, temperature, viscosity and heat conductivity. There are phases -- the subscript 'g' stands for the gas fraction, 'f' -- fluid, 'v' -- vapor. In accord with the inhomogeneous approach the momentum and energy equations source terms should be determined. The volumetric force that acts upon -phase from other phases and the heat flux to -phase can be expressed as [5, 6]: (3) (4) where is mean density, is mean thermal conductivity, is contact area between phases per unit volume. The drag coefficient and Nusselt number are calculated as , (5) , (6) , (7) where is character length-scale of dispersion. Unfortunately this approach does not provide this length-scale (it provides only the volume fraction of each phase), which is an empirical constant of the mode. The multiphase Reynolds system with SST turbulence model is integrated numerically similar to the case of single phase flow [5]. The results of vapour and ventilated cavities simulations are presented in the next section. The computations were conducted in the framework of homogenous and inhomogeneous methods. 1.1. Vapour cavity modeling Consider a water flow with vapour cavity, which is formed by a flat disk-cavitator of diameter mm. The flow velocities are m/s 70m/s, 80m/s, 100m/s.
The flow is treated as isothermal with temperature C. The hydrostatic pressure is Pa, and the saturated vapor pressure is Pa. The computational domain is a sector with 4-degree angle (Fig. 1). Fig.1: Computational domain sketch Each flow case was computed as homogenous and inhomogeneous. The boundary conditions are: - No slip condition on the cavitator surface - Water velocity and volume fraction on the inflow boundary - Hydrostatic pressure on the outflow boundary A structured computational grid consists of 200000 hex elements. Computations of unsteady flow are performed with the time step . The maximum Courant number is less than 40, its average value is about 1. Figure 2a and 2b show distributions of inside and outside cavity computed using the homogenous and inhomogeneous methods, where is dynamic pressure. The water velocity is m/s. Fig. 2a: Distribution of , homogeneous flow Fig. 2b: Distribution of , inhomogeneous flow Figure 3a and 3b show the vapour volume fraction distributions. The blue color corresponds to 100% water, and the red color -- to 100% vapour. Fig. 3a: Vapour volume fraction distribution, homogeneous flow Fig. 3b: Vapour volume fraction distribution, inhomogeneous flow It is seen that the homogeneous and inhomogeneous approaches give essentially different results. The re-entrained Efros' jet predicted theoretically and observed in experiments is modeled in the inhomogeneous computations and it is not modeled by the homogenous approach. This is also observed at other flow velocities. The results of computations are summarized in Table 1 containing theoretical and numerical values of the drag coefficient, , the middle section radius, , and cavity half-length, . The and are modeled rather well in both homogenous and inhomogeneous approach as contrasted to that is quite different from the theoretical data. Table 1 m/s Theoretical Numerical Approach Numerical 50 0.08 0.885 3.4 23 Homog. 0.882 3.4 17 Inhomog. 0.889 3.5 24 70 0.041 0.851 4.7 45 Homog. 0.850 4.7 44 Inhomog. 0.855 4.8 57 80 0.031 0.846 5.3 59 Homog. 0.845 5.2 54 Inhomog. 0.846 5.3 65 100 0.020 0.836 6.6 92 Homog. 0.832 6.8 81 Inhomog. 0.837 6.7 86 Although the inhomogeneous method does not predict well the cavity length, it models qualitative features of the flow physics. The homogeneous method does not model the Efros jet. This and some other reasons indicate that the homogeneous method does not provide correct modeling of supercavitational flow. For example, small bubbles in the water flow rises in the low-pressure region that may cause the phase inversion. In this case, vapor in bubbles spreads and accelerates; i.e., the homogeneous conditions are to be dismissed. 1.2. Ventilated cavity modeling Water and its vapour simultaneous flow are computed when vapour cavity is modeled. To model ventilated cavity the non-condensable gas flow is to be added. The heat exchange between water and gas are also to be considered. It is very important to correctly predict both cavity sizes and gas loss from it. The descriptions of method and results of computations are presented below. Consider incompressible water flow with ventilated cavity, which is formed by a flat disk-cavitator of diameter mm. Inflow gas jet impulse is about zero (see Figure 4). Fig.4: Cavitator sketch The flow velocity is m/s, hydrostatic pressure is Pa, Euler number is . The flow is not treated as isothermal. Water, vapour, gas can have different temperatures.
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