Method of matched asymptotic expansions

About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.

Hydrodynamic loads during early stage of flat plate impact onto water surface

Abstract: The hydrodynamic loads during the water entry of a flat plate are investigated. Initially the water is at rest and the plate is floating on the water surface. Then the plate starts suddenly its vertical motion. The analysis is focused on the early stage during which the highest hydrodynamic loads are generated. The liquid is assumed ideal and incompressible; gravity and surface tension effects are not taken into account. The flow generated by the impact is two dimensional and potential. The penetration depth is either a given function of time or calculated by using the equation of the body motion. A theoretical estimate of the loads during the early stage of the water impact is derived with the help of the method of matched asymptotic expansions. The ratio of the plate displacement to the plate half-width plays the role of a small parameter. The second-order uniformly valid solution of the problem is derived. In order to evaluate the hydrodynamic loads, the second-order pressure distribution is asymptotically integrated along the plate. It is shown that the initial asymptotics of the loads involve a logarithmic term and a negative noninteger power of the nondimensional plate displacement, the latter contribution is related to the inner solution. In addition to the theoretical estimate, a numerical model of the unsteady free-surface flow generated by plate impact is developed. The hydrodynamic loads are numerically evaluated and compared to their asymptotic estimates. A fairly good agreement between the theoretical and numerical predictions of the hydrodynamic loads just after the impact has been found. In the case of constant velocity of the body, it is shown that the relative difference between the theoretical and numerical predictions of the hydrodynamic force is less than 5% when the nondimensional plate displacement is one-fifth and rises to 20% when the nondimensional plate displacement is equal to unity. Similar results are found in the free fall case when the comparison is established in terms of hydrodynamic loads. The theoretical and numerical predictions are remarkably close to each other, even for moderate displacements of the plate, if the comparison is established in terms of the entry velocity.

Asymptotic theory of liquid–solid impact

Abstract: The liquid–solid impact problem is analysed with the help of the method of matched asymptotic expansions. This method allows us to estimate the roles of different effects (viscosity of the liquid, surface tension, compressibility, nonlinearity, geometry) on the impact, to distinguish the regions of the flow and the stages of the impact, where and when each of these effects is of major significance, to present a complete picture of the flow, and describe approximately such phenomena as jetting, escape of the shock onto the liquid–free surface and cavitation. Five stages of the impact are distinguished: supersonic stage, transonic stage, subsonic stage, inertia stage and the stage of developed liquid flow. The asymptotic analysis of each stage is based on general principles of hydrodynamics and will be helpful to design experiments on liquid impact and to develop an adequate computational algorithm, as well as to understand the dynamics of the process.

Two‐Point Taylor Expansions of Analytic Functions

Abstract: Taylor expansions of analytic functions are considered with respect to two points Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals The method is also used for obtaining Laurent expansions in two points