Eugeny Buldakov

New asymptotic description of nonlinear water waves in Lagrangian coordinates

A new description of two-dimensional continuous free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in R 2 . Components of the transformation function satisfy the linear Euler-type continuity equation and can be expressed via a single function analogous to an Eulerian stream function. Fixed-point iterations lead to a simple recursive representation of a solution satisfying the Lagrangian continuity equation. Expanding the unknown function in a small-perturbation asymptotic expansion we obtain the complete asymptotic formulation of the problem in a fixed domain of Lagrangian labels. The method is then applied to the classical problem of a regular wave travelling in deep water, and the fifth-order Lagrangian asymptotic solution is constructed, which provides a much better approximation of steep waves than the corresponding Eulerian Stokes expansion. In contrast with early attempts at Lagrangian regular-wave expansions, the asymptotic solution presented is uniformly valid at large times.

On transonic viscous-inviscid interaction

The paper is concerned with the interaction between the boundary layer on a smooth body surface and the outer inviscid compressible flow in the vicinity of a sonic point. First, a family of local self-similar solutions of the Karman–Guderley equation describing the inviscid flow behaviour immediately outside the interaction region is analysed; one of them was found to be suitable for describing the boundary-layer separation. In this solution the pressure has a singularity at the sonic point with the pressure gradient on the body surface being inversely proportional to the cubic root dpw/dx [similar] ([minus sign]x)[minus sign]1/3 of the distance ([minus sign]x) from the sonic point. This pressure gradient causes the boundary layer to interact with the inviscid part of the flow. It is interesting that the skin friction in the boundary layer upstream of the interaction region shows a characteristic logarithmic decay which determines an unusual behaviour of the flow inside the interaction region. This region has a conventional triple-deck structure. To study the interactive flow one has to solve simultaneously the Prandtl boundary-layer equations in the lower deck which occupies a thin viscous sublayer near the body surface and the Karman–Guderley equations for the upper deck situated in the inviscid flow outside the boundary layer. In this paper a numerical solution of the interaction problem is constructed for the case when the separation region is entirely contained within the viscous sublayer and the inviscid part of the flow remains marginally supersonic. The solution proves to be non-unique, revealing a hysteresis character of the flow in the interaction region.

On the uniqueness of steady flow past a rotating cylinder with suction

The subject of this study is a steady two-dimensional incompressible flow past a rapidly rotating cylinder with suction. The rotation velocity is assumed to be large enough compared with the cross-flow velocity at innity to ensure that there is no separation. High-Reynolds-number asymptotic analysis of incompressible Navier{ Stokes equations is performed. Prandtl’s classical approach of subdividing the flow eld into two regions, the outer inviscid region and the boundary layer, was used earlier by Glauert (1957) for analysis of a similar flow without suction. Glauert found that the periodicity of the boundary layer allows the velocity circulation around the cylinder to be found uniquely. In the present study it is shown that the periodicity condition does not give a unique solution for suction velocity much greater than 1=Re. It is found that these non-unique solutions correspond to dierent exponentially small upstream vorticity levels, which cannot be distinguished from zero when considering terms of only a few powers in a large Reynolds number asymptotic expansion. Unique solutions are constructed for suction of order unity, 1=Re, and 1= p Re. In the last case an explicit analysis of the distribution of exponentially small vorticity outside the boundary layer was carried out.

The Euler-type Description OfLagrangian Water Waves

A new description of 2D continuous free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in R. Components of a transformation function satisfy the linear Euler-type continuity equation and can be expressed via a single function analogous to an Eulerian stream function. Fixed-point iterations lead to a simple recursive representation of a solution satisfying the Lagrangian continuity equation. Expanding the unknown function into a small-perturbation asymptotic expansion we obtain the complete asymptotic formulation of the problem in a fixed domain of Lagrangian labels. The method is then applied to a classical problem of a regular wave traveling in deep water, and the fifth order Lagrangian asymptotic solution is constructed. In contrast with early attempts of Lagrangian regular-wave expansions, the presented asymptotic solution is uniformly-valid at large times.

The Effect of Directional Spreading on Interaction of Focused Wave Groups With a Cylinder

The problem of diffraction of a directionally spread focused wave group by a bottom-seated circular cylinder is considered from the view point of second-order perturbation theory. After applying the time Fourier transform and separation of vertical variable the resulting two-dimensional non-homogeneous Helmholtz equations are solved numerically using finite differences. Numerical solutions of the problem are obtained for JONSWAP amplitude spectra for the incoming wave group with various types of directional spreading. The results are compared with the corresponding results for a unidirectional wave group of the same amplitude spectrum. Finally we discuss the applicability of the averaged spreading angle concept for practical applications.

Viscous-Inviscid Interaction and Boundary-Layer Separation in Transonic Flows

In this paper theoretical analysis of transonic flow separation from a rigid body surface is presented. Two forms of separation are considered. The first one is observed when the boundary layer separates from a corner point of a rigid body contour, say, an aerofoil. The second takes place on a smooth part of the surface. For both cases the flow behaviour is studied based on the asymptotic analysis of the Navier-Stokes equations. In this analysis the Reynolds number is assumed large, and the Mach number of the inviscid flow at the separation point is close to one. We found that the flow separating from a corner is driven towards the separation by inviscid-inviscid interaction between the boundary layer and inviscid external flow. Meanwhile separation on a smooth surface is accompanied by a more traditional viscous-inviscid separation. However, unlike in subsonic or supersonic flow, the boundary layer immediately upstream of the interaction region has a preseparated form. This results in a hysteresis character of the flow behaviour in the interaction region.

Certain features of similarity solutions for flows in viscous vortex cores

A class of steady similarity solutions of the equations for viscous vortex cores which correspond to external inviscid similarity solutions with a power-law variation of the circumferential velocityv-r−m near the rotation axis is considered. It is found that if the Bernoulli function in external flow is constant, then these solutions will exist only on a certain range of the indexm of the exponential. For eachm on this range there are two solutions.

Properties of similarity solutions for flows in turbulent vortex cores

A class of similarity solutions of the equations for turbulent vortex cores matching an external inviscid similarity flow with a power law of circumferential velocity variationv-r−m near the rotation axis and constant Bernoulli function is considered. Solutions are found to exist only in a certain range of the indexm of the exponential. For each suchm there are two solutions.