Parabolic Boundary Layer Research Articles

Electrically Driven Flows in MHD with Mixed Electromagnetic Boundary Conditions

AbstractFlow of viscous, incompressible, electrically conducting fluid, driven by imposed electric currents has been investigated in the presence of a transverse magnetic field. The boundary perpendicular to the magnetic field is perfectly conducting partly along its length. Three cases have been considered: a) flow in the upper half plane when the boundary to the right of origin is insulating and that to the left is perfectly conducting, b) flow in the upper half plane when a finite length of the boundary is perfectly conducting, and c) flow in a flat channel when a finite length of boundary in each plane, symmetrically situated, is perfectly conducting. In case a), an exact analytical solution is derived, from which the existence of a boundary layer, parabolic in shape and emanating from the point of discontinuity in electrical boundary conditions, is established. In cases b) and c), the problem is reduced analytically to a Fredholm's integral equation of second kind, which is solved numerically. A number of transformations and valid approximations are used to simplify the task of numerical computation. Velocity contours and current lines have been plotted for each problem. For large values of Hartmann number the parabolic boundary layer mentioned above is demarcated.

Asymptotic Analysis of a Singular Perturbation Problem

We study, in a rectangle $0 < x < a$ and $0 < y < b$, the Dirichlet problem for an elliptic differential equation of the form \[ - \epsilon \Delta u\epsilon + p\frac{{\partial u\epsilon }}{{\partial x}} + qu\epsilon = f(x,y)\] where $\epsilon $ is a small parameter $0 < \epsilon \ll 1$, $\Delta $ is the Laplacian operator, p is a positive number, q is a nonnegative number and all of the input data are smooth. We establish a constructive procedure for obtaining an asymptotic approximation of arbitrary order with respect to $\epsilon $ of this singular perturbation problem, and also give a proof of its uniform validity in the closed rectangle by the use of the maximum principle and exponential estimates of all boundary or corner layer functions. The corner singularities of parabolic boundary layer functions are removed by introducing elliptic boundary layer functions along the characteristic boundaries $y = 0$ and $y = b$. Both ordinary corner layer functions and elliptic corner layer functions are employed at the outflow corners $(a,0)$ and $(a,b)$. An application is made to settle a long-standing problem in the magnetohydrodynamic flow in a rectangular duct.

Allowance for longitudinal diffusion in the neighborhood of a discontinuity of catalytic surface properties

In the investigation of flow near surfaces with discontinuous changes in the catalytic properties the question arises of the applicability of parabolic boundary and viscous shock layer equations in the neighborhood of the discontinuity. In the present paper, three types of problem are solved in which longitudinal diffusion is taken into account. In the first an insertion with different catalytic properties is placed in the neighborhood of the stagnation point, in the second the discontinuity lines of the catalytic properties are perpendicular to the oncoming flow, while in the third they are parallel. On the main surface and on the insertion surface the heterogeneous catalytic reactions are assumed firstorder reactions with various rate constants whose values vary in a wide range. The data of the solution are compared with the solution obtained using the boundary layer approximation and the regions of influence of the longitudinal diffusion are estimated. In [1–4] a problem similar to the second one was solved by the numerical method of [1] and the Wiener-Hopf method for the case of transition from a noncatalytic to a perfectly catalytic surface and the region of applicability of the boundary layer was estimated [5].

Necessary convergence conditions for upwind schemes in the two‐dimensional case

AbstractThis paper considers nine‐point difference schemes for a two‐dimensional boundary value singular perturbation problem without turning points and parabolic boundary layers. Necessary conditions are given for the uniform convergence (in the sense of the maximum norm) of a scheme. Using these conditions, several widely used schemes are analysed. It is shown that some common schemes are not uniformly convergent in ϵ. and that in some cases we are able to compute uniquely free parameters in the scheme. Some remarks on the treatment of a problem with a parabolic boundary layer are given.

A theoretical model for predicting the performance of circulation controlled aerofoils and cascades

SummaryThe procedure described applies to aerofoils or cascades with circulation controlled by tangential blowing jets and provides a complete numerical solution for aerodynamic performance in incompressible flow conditions.The pressure distribution over the blade surface is predicted by a potential flow model in which the region of separated flow is represented by an appropriated source distribution. Boundary layer development is calculated by a finite-difference solution of the parabolic boundary layer momentum equation: the development of the blowing jet and its mixing with the boundary layer over the curved trailing edge is predicted by the same procedure applied to an angular momentum equation, using an intermittency representation of the eddy viscosity distribution.Predictions are compared with experimentally measured lift coefficients for an isolated aerofoil, with turning angles for a cascade tested by other workers and with experimental turning angles for a cascade tested by the authors. The agreement between theory and experiment is good.

On three-dimensional boundary layer flow over surface irregularities

The three-dimensional pipeflow boundary layer equations of Smith (1976) are shown to apply to certain external flow problems, and a numerical method for their solution is developed. The method is used to study flow over surface irregularities, and some three-dimensional separated flows are calculated. Upstream influence in the form of so-called ‘free interactions’ requires an iterative solution technique, in which the initial conditions for the parabolic boundary layer equations must be determined to satisfy a downstream condition

Some comments on a singular elliptic problem associated with a rectangle

SynopsisThe construction of the approximate solution within a rectangle of a singular elliptic problem is discussed. It is found that, provided the boundary data satisfy certain continuity conditions at the corners of the rectangle, ordinary boundary layers and parabolic boundary layers only are necessary to describe the solution. A correction term, however, has to be added to the solution if the continuity conditions on the boundary data are not satisfied.

The Behavior as $\varepsilon \to 0^ + $ of Solutions to $\varepsilon \nabla ^2 w = ({\partial / {\partial y}})w$ on the Rectangle $0 \leqq x \leqq l$,$| y | \leqq 1$

The title problem is first examined in the limit of the semi-infinite strip $l = \infty $, for boundary data $w(x, - 1) = f(x)$, $w(x,1) = g(x)$, $w(0,y) = h(y)$. Here f, g, h are infinitely differentiable except at the corners where one-sided derivatives of all orders exist. Previous work on the infinite strip covers cases where $h = 0$ so that (by superposition) the present discussion may be narrowed to cases where $f = g = 0$; for these the solution is asymptotically zero for $x \geqq x_0 > 0$. Near $x = 0$ four regions are distinguished: the parabolic boundary layer $y \leqq y_1 0$ and the transition zone both of which are determined by the parabolic layer. By means of Fourier sine...

Velocity and temperature profiles in compressible turbulent wall jets

AbstractThe results of an experimental study on compressible turbulent wall jets resulting from a normally impinging round jet are presented. A finite difference technique was used for the calculation of velocity and temperature profiles. The eddy transfer coefficients used were functions of the distance from the plate throughout the flow field of the wall jet. The nozzle exit Mach number ranged up to 0.85. The nozzle exit temperature to the ambient temperature ratio ranged up to 2.92. The applicability of the calculational method has been thoroughly checked for a region of flow extending up to 12 nozzle diameters from the axis of symmetry. Using the empirical eddy transfer coefficients presented in this paper only starting profiles are needed to generate the solution for any point in the wall jet flow field which is described by a system of parabolic (boundary layer) equations.

On singular perturbations and parabolic boundary layers

A singular perturbation problem involving parabolic boundary layers is investigated. The second approximation for the boundary layer is constructed and it is shown that this approximation cannot be obtained by the usual perturbation method.

Simplified Analysis of Boundary-Layer Oscillations

The stability of an incompressible boundary layer is analyzed in terms of three basic processes. These are (a) the oscillations of a boundary layer when friction is disregarded, (b) the effects of friction at the wall, and (c) the effects of friction at the critical layer. These processes are separately discussed and evaluated. Simple models are presented. A general equation leads to the eigenvalues. The neutral curves corresponding to five typical cases are determined—parabolic and Blasius boundary layers, boundary layers with suction and with adverse pressure gradient, two-dimensional Poiseuille flow. The unstable boundary layer is discussed briefly. The nonlinear effects of the oscillation on the velocity profile are evaluated. Finally, the case of a boundary layer along an elastic wall is considered, and it is found that the wall may have a significant effect on the layer. In particular, a wall with negative damping could completely stabilize the boundary layer.