The Three-Dimensional Neumann Problem with Generalized Boundary Conditions and the Prandtl Equation

Setukha, A. V.

doi:10.1023/b:dieq.0000012692.94617.c1

Cited by 9 publications

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“…Poltavskii for the equation (5) in the case plane open surface and in the case of the regular partition. The convergence of approximate solution to the exact solution in the sense of C-norm on the grid was obtained in the work [7] for the equation (5) and in the work [5] for the equation (6) Note, that the similar numerical scheme for the equation (4) in the case of closed surface and k = 0 (the Laplace equations) with smoothing of the singularity in the integral was constructed in [6] (the ideas of Ryzhakov G.V.) …”

Section: Methodsmentioning

confidence: 99%

The singular integral equation method in 3-D boundary value problems and its applications

Setukha¹

2012

AIP Conference Proceedings

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The integral equations with hypersingular integrals on closed and opened surfaces which appear when finding the solution of the 3-dimensional Neumann boundary value problem for Laplace or Helmholtz equations in double-layer potential form are considered in the paper. Author presents his results regarding the solvability of equations of this type, the convergence of the numerical method of their solution like method of discrete singularities in uniform metric on a grid. Also examples of the method practical implementation in applied problems of aerodynamics and acoustics are given.

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Section: Methodsmentioning

confidence: 99%

The singular integral equation method in 3-D boundary value problems and its applications

Setukha¹

2012

AIP Conference Proceedings

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“…On the basis of the theory extended in [4] the authors prove that the formulated problem is a solvable one and the physical conditions given in Subsection 2.1 for the corresponding velocity eld are satis ed. Thus, if we consider a ow past the wing with ow suction at the point q 2 S, the velocity vector ux through any part of the surface S, which comprises the point q, on the positive side is equal to the suction intensity Q.…”

Section: Reducing the Problem Of Ow Past A Wing With Ow Suction To Thmentioning

confidence: 99%

“…Following [4] we introduce on the surface S the generalized functions as continuous linear functionals over the spaces of the main functions, which are introduced in a special way, i.e. in the sense of Sobolev-Schwartz.…”

Section: Reducing the Problem Of Ow Past A Wing With Ow Suction To Thmentioning

confidence: 99%

“…We use the Neumann boundary value problem theory with generalized boundary conditions, which was extended in [4]. The proposed method is a further development of the discrete vortex frame method that allows one to model a ow past bodies and the evolution of a vortex wake behind them in the framework of an ideal incompressible uid model.…”

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“…We use the Neumann boundary value problem theory with generalized boundary conditions, which was extended in [4]. We give the physical formulation of the problem assuming that the normal component of the uid velocity on the wing surface must be zero everywhere beyond the suction point (suction line), while the uid ow through the wing surface on the side of suction is nonzero.…”

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On numerical modelling of a three-dimensional flow past a wing with external flow suction and on the effect of flow suction on trailing vortices

Dimitroglo

Lifanov

Setukha

2004

Russian Journal of Numerical Analysis and Mathematical Modelling

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In this paper, we propose a numerical method of calculating a three-dimensionalunsteady ow past a thin wing in the presence of ow suction on the wing surface through a point hole or a narrow slot. The proposed method is a further development of the discrete vortex frame method that allows one to model a ow past bodies and the evolution of a vortex wake behind them in the framework of an ideal incompressible uid model.When developing a mathematical model that takes into account the ow suction from the wing surface we rst consider the problem of a potential steady ow past a wing (without a vortex wake). We give the physical formulation of the problem assuming that the normal component of the uid velocity on the wing surface must be zero everywhere beyond the suction point (suction line), while the uid ow through the wing surface on the side of suction is nonzero. To satisfy this physical condition we propose to pose a mathematical boundary value problem for the velocity eld potential in which the normal derivative of an unknown function on one of the sides of the surface is a delta function concentrated at the suction point (on the suction line). We use the Neumann boundary value problem theory with generalized boundary conditions, which was extended in [4]. On the basis of this theory, we prove the solvability of the posed boundary value problem, investigate the behaviour of the arising velocity eld in a neighbourhood of the suction point, propose and justify the numerical method of solving the problem. Further the developed method of solving the stationary problem is empirically coupled with the known method of the numerical solution to the unstationary problem of ow past a wing with vortex wake evolution.Using the developed mathematical model, we numerically modelled a ow past a rectangular wing in the presence of ow suction through a narrow slot or a point hole. We investigated the vortex wake structure at various positions of a ow suction device. In the numerical modelling it was found that at certain positions of ow suction devices there is a loss of the stability of trailing vortices and their destuction process begins at distances from a wing, which are several times smaller than those without suction.¤ N. E. Zhukovsky Air Force Engineering Academy,

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On the Three-Dimensional Neumann Boundary Value Problem with a Generalized Boundary Condition in a Domain with Smooth Closed Boundary

Setukha

2005

Diff Equat

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The present paper continues the study of the Neumann boundary value problem for the Laplace equation with a generalized boundary condition. This study was initiated in [1][2][3][4], where a screen problem was considered: the solution is sought outside a thin surface (screen) and the boundary condition is posed on both sides of the surface; moreover, the surface was assumed to be a part of the plane. The present paper deals with the case of interior and exterior problems in domains bounded by a smooth closed surface. The essential new feature is that the surface is not plane.We assume that the right-hand side of the boundary condition is a distribution treated as a continuous linear functional on the space of some test functions. We introduce the notion of generalized boundary values and generalized normal derivative of a function on the boundary of the domain. We prove the uniqueness of the solution of the problem. The existence of the solution is proved for the case in which the right-hand side of the boundary conditions is a continuous linear functional on the space of continuous functions on the boundary of the domain; moreover, the solution is constructed as a superposition of elements of a system of fundamental solutions.The present paper uses the notation introduced in [4].

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The Three-Dimensional Neumann Problem with Generalized Boundary Conditions and the Prandtl Equation

Cited by 9 publications

References 0 publications

The singular integral equation method in 3-D boundary value problems and its applications

The singular integral equation method in 3-D boundary value problems and its applications

On numerical modelling of a three-dimensional flow past a wing with external flow suction and on the effect of flow suction on trailing vortices

On the Three-Dimensional Neumann Boundary Value Problem with a Generalized Boundary Condition in a Domain with Smooth Closed Boundary

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