Method for solving the equations describing the triple-deck interaction of a three-dimensional boundary layer with an outer inviscid transonic flow

An approach is developed for solving boundary value problems describing three-dimensional steady flows in the region where a laminar boundary layer interacts with an outer inviscid transonic flow. By applying the method, the flow over a roughness element is computed within the classical triple-deck theory, the asymptotic height of the roughness element corresponding to nonseparated flow is determined, and separated flow patterns are constructed.     

The boundary layer on a wall nearly parallel to a magnetic field

This paper calculates the structure of the boundary layer which occurs at the wall of a nonconducting duct in fully-developed laminar flow of a conducting fluid when the wall is inclined at a small angle to the uniform transverse magnetic field and when the Hartmann numberM is high. When is large compared withM ^–1/2 the layer splits into a Hartmann layer on the inclined wall and a weak shear layer emanating from the duct corner along field lines. These two layers cause equal reductions in the total flow rate.

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Zusammenfassung Diese Arbeit behandelt die Struktur der Grenzschicht, die an der Wand eines nicht leitfähigen Kanals entsteht die eine vollentwickelte Laminarströmung einer leitfähigen Flüssigkeit führt, wobei die Wand einen kleinen Winkel mit dem gleichförmigen transversalen Magnetfeld bildet und die Hartmannsche ZahlM groß ist. Für große Werte von im Vergleich zuM ^–1/2 teilt sich die Grenzschicht in einen Hartmannsche Schicht an der geneigten Wand und eine schwache Scherschicht, die von der Ecke des Kanals entlang der Feldlinien ausgeht. Die beiden Schichten verursachen die gleichen Verminderungen in der Durchflußmenge.

     

Sequences of functionals with a boundary layer

Siberian Mathematical Journal -     

Singularly perturbed boundary value problems with a power-law boundary layer

We consider boundary value problems for second-order singularly perturbed equations whose solution has a power-law boundary layer that occurs because the degenerate equation has multiple roots.     

Singular integrals with generalized Cauchy kernels for the velocity field in boundary value problems of filtration in an anisotropic inhomogeneous porous layer

We study two-dimensional stationary and nonstationary boundary value problems of fluid filtration in an anisotropic inhomogeneous porous layer whose conductivity is modeled by a not necessarily symmetric tensor. For the velocity field, we introduce generalized singular Cauchy and Cauchy type integrals whose kernels are expressed via the leading solutions of the main equations and have a hydrodynamic interpretation. We obtain the limit values of a Cauchy type generalized integral (Sokhotskii-Plemelj generalized formulas). This permits one to develop a method for solving boundary value problems for the filtration velocity field. The idea of the method and its efficiency are illustrated for the boundary value problem of filtration in adjacent layers of distinct conductivities and the problem of the evolution of liquid interface.     

Convergence proof of the velocity field for a stokes flow immersed boundary method

The immersed boundary (IB) method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures. It is characterized by the use of a uniform Cartesian mesh for the fluid, a Lagrangian curvilinear mesh on the elastic material, and discrete delta functions for communication between the two grids. We consider a simple IB problem in a two‐dimensional periodic fluid domain with a one‐dimensional force generator. We obtain error estimates for the velocity field of the IB solution for the stationary Stokes problem. We use this result to prove convergence of a simple small‐amplitude dynamic problem. We test our error estimates against computational experiments. © 2007 Wiley Periodicals, Inc.     

Solutions of MHD boundary layer equations with transverse magnetic field and arbitrary pressure gradient

Certain solutions of Magnetohydrodynamic boundary layer equations for a flat plate with a transverse magnetic field fixed relative to the fluid have been obtained using a power-series method given by Hassan. This power series solution has all the desirable qualities of Görtler series and, in addition, its zeroth order term which is governed by a non-linear total differential equation can be given in closed form. The first order term governed by a linear total differential equation has been integrated numerically. The results are tabulated for various values of S, the interaction parameter. The results show a decrease in the boundary layer thickness with a consequent increase in skin friction as the strength of magnetic field is increased.     

Control of a boundary layer separation

The results of experimental study on a boundary layer separation control are given in the paper. The boundary layer on a flat wall is subjected to adverse pressure gradient. The active control strategy evolving a synthetic jet has been applied. The separation process is investigated using TR-PIV method. Dynamical aspects of the phenomenon are analyzed in details. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrable hydrodynamic submodels with a linear velocity field

Invariant and partially invariant solutions to the equations of gas dynamics with a linear velocity field are defined by a matrix satisfying a homogeneous integrable Riccati equation. The classification is carried out of solutions by the acceleration vector in the Lagrangian coordinates. Some example is given of an invariant solution for which the selected volume “collapses” to an interval.     

Discrete velocity models with general boundary conditions in a slab

We consider a gas between parallel plates, described by a discrete velocity model. At the boundary, we impose the most general linear boundary conditions which preserve mass. Using a fixed‐point theorem we prove the existence of at least one parameter family of solutions, continuous in x. The velocities are assumed to have a non‐zero component in the direction orthogonal to the boundaries. Copyright © 2001 John Wiley & Sons, Ltd.     

Nano boundary layer equation with nonlinear Navier boundary condition

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.     

The effect of suction on boundary layer for rotating flows with or without magnetic field

The effect of suction on the steady laminar incompressible boundarylayer flow for a stationary infinite disc with or without magnetic field, when the fluid at a large distance from the surface of the disc undergoes a solid body rotation, has been studied. The governing coupled nonlinear equations have been solved numerically using the shooting method with least square convergence criterion. It has been found that suction tends to reduce the velocity overshoot and damp the oscillation.     

The effects of magnetic field oscillations on the boundary layer flow past a magnetized plate

The effects of harmonic oscillations of amplitude in the magnitude of the magnetic field on the steady boundary layer flowO( ²) due to a uniform free stream past a semi-infinite magnetized flat plate have been discussed. It is observed that the magnetic field distribution does not affect the surface current distribution.

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Zusammenfassung Die Effekte einer harmonischen Schwingung der Amplitude im Betrag des Magnetfeldes auf die Grenzschicht entlang einer halb-unendlichen magnetischen Platte in gleichförmiger Strömung wird diskutiert. Man findet, daß die magnetische Feldverteilung die Oberflächenströme nicht beeinflußt.

     

On a double boundary layer in a nonlinear boundary value problem

The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than 2 are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than 2 such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches 1 monotonically at each point in the source domain.     

Integral boundary problem in a layer

Translated from Matematicheskie Zametki, Vol. 53, No. 6, pp. 122–129, June, 1993.     

Mathematical model of a kinetic boundary layer

The two-dimensional (plane) problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a homogeneous polyatomic gas flow with no dissociation or electron excitation is considered assuming that energy exchange between translational and internal molecular degrees of freedom is easy. (The approximation of a hypersonic kinetic boundary layer arises from the kinetic theory of gases and, within the thin-layer model, takes into account the strong nonequilibrium of the hypersonic flow with respect to translational and internal degrees of freedom of the gas particles.) A method is proposed for constructing the solution of the given kinetic problem in terms of a given solution of an equivalent well-studied classical Navier-Stokes hypersonic boundary layer problem (which is traditionally formulated on the basis of the Navier-Stokes equations).