Background

The impact of a body entering a nearly parallel fluid region has a wide range of applications, ranging from ship slamming and inkjet printing, to asteroid impact. The principal aim of this project is to study recent and ongoing experiments by Prof. S.T. Thoroddsen (Mechanical Engineering, KAUST) to help develop new mathematical models for splash dynamics. In particular, we are considering extending the existing theory to oblique impact problems and modelling the influence of air cushioning on splash jets.

Techniques and Challenges

The general water-entry model, even in its simplest form, is highly nonlinear. We use the assumption of small deadrise, so that the impactor is nearly flat, to make progress using the method of matched asymptotic expansions. In particular, we have utilised a variational transformation to rewrite the oblique water-entry problem in terms of its normal impact counterpart. We are also working on including air cushioning in the leading-order problems, although analytical progress of the resulting system of equations becomes much more difficult. In terms of jet dynamics, the difficulties lie in understanding the parameter regimes in which various splash phenomena may be realised. We are developing a hierarchy of models to describe the evolution of splash jets in different regimes.

Results

Utilising the variational form, we have managed to solve the oblique water-entry problem by reformulating it as the corresponding normal impact problem. In particular, we have solved the general symmetric two-dimensional and axisymmetric problems and considered the breakdown of the theory and ideas of cavitation. Furthermore, we have developed a model for oblique water-entry incorporating air-cushioning at leading order. The most general form of the splash jet equations has been derived.

The Future

We will develop a hierarchy of models for the splash jet and investigate its stability and behaviour. In particular, we will cooperate with Prof. S.T. Thoroddsen in using experimental data to inspire the various parameter regimes we consider and also to compare analytical and numerical results.

References

[12/32] Moore M.R., Howison S.D., Ockendon J.R., Oliver J.M.: Three-dimensional oblique water-entry problems at small deadrise angles, J. Fluid Mech.

[12/18] Moore M.R., Howison S.D., Ockendon J.R., Oliver J.M.: A note on oblique water entry, J. Engg. Math.

Howison S.D., Ockendon J.R., Wilson S.K.: Incompressible water-entry problems at small deadrise angles, J. Fluid Mech. 222, pp. 215-230, 1991

Howison S.D., Ockendon J.R., Oliver J.M.: Oblique slamming, planing and skimming, J. Engg. Math. 48 pp. 321-337, 2004

Thoroddsen S.T., Thoraval M.J., Takehara K., Etoh T.G.: Droplet splashing by a slingshot mechanism, Physical Review Letters, 106(3), 2011