JETTING BY FLOATING WEDGE IMPACT

The problem of sudden vertical motion of a floating wedge was studied by Iafrati & Korobkin (2003) within the potential theory of incompressible liquid flow. Combined initial asymptotics of the generated flow has been derived and analyzed in details. The flow region was divided into the main flow region and small vicinities of the intersection points between the wedge side walls and the liquid free surface. The inner flow close to the intersection points was obtained by means of combination of numerical and analytical methods. It was shown that the inner flow is non-linear and self-similar in the leading order during the initial stage of the impact. Close to the intersection points the free surface is turned over and the jets are originated. It should be noted that the free surface shape was numerically calculated as a part of the solution. The analysis of the non-linear boundary-value problem in the inner region revealed that this problem cannot describe the flow in the jet region and, in particular, cannot provide estimate of the jet length at the initial stage of the impact. Asymptotic analysis of the inner solution showed that the jet is predicted to be of infinite length with the jet thickness rapidly decreasing with the distance from the intersection points but with the flow velocity in the jet growing linearly with the distance. This implies that the first-order solution derived by Iafrati & Korobkin (2003) is not uniformly valid and should be improved in the jet region. The flow in the jet region caused by a floating body impact was not studied before. We do not expect that the details of the flow in thin jet region strongly affect either the pressure distribution in the main flow region or the free surface shape outside the jet region. Moreover, these thin jets are expected to be disintegrated into clouds of droplets owing to instability of the high-speed jets and, therefore, cannot be detected in experiments. However, formally speaking, the analysis by Iafrati & Korobkin (2003), which does not account for instability mechanisms, is incomplete because it does not contain explanations of the origin of the jets and gives no idea how the jet length can be estimated, which makes confusions in practical implications of the derived initial asymptotics. These subjects are covered in the present report. It should be noted that infinite jets within the incompressible liquid model are well known in the theory of water entry problems (Wilson, 1989). Due to physical reasons - the liquid particles in the jets move inertially and independently, the feedback of the jet flow to the flow in the main region can be well neglected - the jet flow was not studied for long time. Analysis of such jets was initiated by attempts to calculate kinetic energy evacuated from the main flow region with these jets (Korobkin, 1994). Surprisingly, it was found that the jet energy is comparable with that in the main bulk of the liquid. Later on, the better method of estimating the jet energy was suggested (Molin el al., 1996), which does not require details of the flow in the jet region. Nevertheless, in the problem of blunt body impact onto a liquid free surface Korobkin (1997) argued that the compressibility effects should be taken into account to obtain both the shape and the length of the spray jet. Calculations have been performed for the entry of a parabolic contour at constant velocity. This idea is used in the present report to estimate the length of the jet produced by impact of a floating wedge. The jetting by a floating wedge impact starts at the very early stage, when the compressibility effects must be taken into account and the disturbed part of the liquid is localized near the body surface. In Figure 1 the wave pat- tern at this stage of the impact is shown: BF and B'F' are the shock fronts, CFA, C'F'A' and E'B'GBE are the fronts of the relief waves. If the impact velocity V is much smaller than the sound speed c_0 in the liquid at rest, the fronts of the relief waves are approximately circular with the radius being equal to c_0 t , where t is dimensional time. The distinguished stage lasts until points C and E meet each other, which happens at t = t* , t* = |DC|/(2c_0 ). During this stage the free surface is already deformed and the jet region is formed (these deformations are not shown in the figure). The liquid particles, which entered the jet region during this stage, will form the jet head at any following time instant. Important feature of the stage is that the flows in regions BFCE and B'F'C'E' are one-dimensional and the flows in regions CFA, C'F'A' and E'B'GBE are self-similar, which highly simplifies the analysis. Within the acoustic approximation the pressure in BFCE is constant p'= ?_l0 c_0 V_i and the flow velocity is equal to the normal velocity of the wedge surface V_i = V cos ?, ?_l0 is the density of the liquid at rest and ? is the wedge deadrise angle. The flow in E'B'GBE is not considered here. We consider the flow in CFA, where the jet is initiated, within the moving coordinate system xPy attached to the wedge wall DP. Both the flow and wave patterns are shown in Figure 2. In the moving coordinates the flow is equivalent to that due to the liquid wedge impact onto the rigid plate and is self-similar. We shall determine the uniformly valid asymptotics of the liquid flow and the pressure distribution in the region CFALP with the Mach number M = V_i /c_0 being a small parameter of the problem, and evaluate the jet length L_jet = |P L|