Generalised Wagner model of water impact by numerical conformal mapping

• The original problem is reduced to two non-linear integral equations, where one of them serves to evaluate the conformal mapping of the region and the other one to compute the position of the contact point.
• Both equations are singular but their solutions are obtained with good accuracy.
• The obtained solutions predict accurately the hydrodynamic forces for blunt sections.
• Calculations of the forces and the pressures by the present method are quick after precalculations dependent only on the body shape.

A numerical method to solve the problem of symmetric rigid contour entering water vertically at a given time-dependent speed is presented. The method is based upon the so-called generalised Wagner model. Within this model the body boundary condition is imposed on the actual position of the entering surface, the free-surface boundary conditions are linearised and imposed on the pile-up height, which is determined as part of the solution. The hydrodynamic pressure is given by the non-linear Bernoulli equation. The hydrodynamic pressures which are below the atmospheric value are disregarded. The conformal mapping of the flow region onto the lower half-plane is used. The velocity potential of the flow is given in analytical form once this mapping is known. The conformal mapping is calculated numerically. The obtained results are validated with respect to the known solutions for wedge and circular cylinder. The novelty and practical importance of the present approach are due to a special accurate treatment of the flows and the pressures close to the contact points between the entering body and water free surface. This special treatment is required for reliable prediction of the hydrodynamic pressure along the wetted part of the contour during its impact onto the water surface and the subsequent entry.