Further Solution of the Corner Boundary-Layer Equations

1973 ◽  
Vol 24 (3) ◽  
pp. 219-226 ◽  
Author(s):  
M Zamir
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Existence And Uniqueness ◽  
Boundary Layer Equation ◽  
Critical Value ◽  
Numerical Accuracy ◽  
Boundary Layer Equations ◽  
The Past ◽  
Similar Solutions ◽  
High Degree

SummarySimilar solutions of an equation governing the flow in the plane of symmetry of a corner boundary layer with favourable pressure gradient are extended to a “critical” value of the pressure gradient for which no solution could be found previously. It is shown that the failure to achieve this result in the past was due to the “singular” nature of this solution rather than to its non-existence as one was tempted to suspect. The existence and uniqueness of this crucial solution are demonstrated and the solution itself is obtained to a high degree of numerical accuracy. Criticism of the theory on which this corner boundary-layer equation is based is discussed in an Appendix.

The behaviour of similar solutions in a compressible boundary layer

1968 ◽  
Vol 34 (2) ◽  
pp. 337-342 ◽  
Author(s):  
J. B. Mcleod ◽  
J. Serrin
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Compressible Boundary Layer ◽  
Rigorous Proof ◽  
Velocity Overshoot ◽  
Boundary Layer Equations ◽  
Mathematical Properties ◽  
Model Fluid ◽  
Similar Solutions

This paper discusses the mathematical properties of similar solutions of the boundary-layer equations in a compressible model fluid, under assumptions first introduced by Stewartson and by Li & Nagamatsu. Assuming a favourable pressure gradient and that backflow is not present, our results include (among other things) a rigorous proof that velocity overshoot occurs in the boundary layer if the wall is heated, and that this is true whether or not suction, blowing or slipping occurs at the wall; while, conversely, velocity overshoot does not occur when the wall is cooled and the amount of slipping at the wall is suitably restricted.

Separation Criterion for Turbulent Boundary Layers Via Similarity Analysis

2004 ◽  
Vol 126 (3) ◽  
pp. 297-304 ◽  
Author(s):  
Luciano Castillo ◽  
Xia Wang ◽  
William K. George
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Boundary Layers ◽  
Boundary Layer Equation ◽  
Outer Part ◽  
Adverse Pressure Gradient ◽  
Gradient Flows ◽  
Boundary Layer Equations ◽  
Gradient Parameter ◽  
Momentum Boundary Layer

By using the RANS boundary layer equations, it will be shown that the outer part of an adverse pressure gradient turbulent boundary layer tends to remain in equilibrium similarity, even near and past separation. Such boundary layers are characterized by a single and constant pressure gradient parameter, Λ, and its value appears to be the same for all adverse pressure gradient flows, including those with eventual separation. Also it appears from the experimental data that the pressure gradient parameter, Λθ, is also approximately constant and given by Λθ=0.21±0.01. Using this and the integral momentum boundary layer equation, it is possible to show that the shape factor at separation also has to within the experimental uncertainty a single value: Hsep≅2.76±0.23. Furthermore, the conditions for equilibrium similarity and the value of Hsep are shown to be in reasonable agreement with a variety of experimental estimates, as well as the predictions from some other investigators.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

scholarly journals Similarity Solutions of the MHD Boundary Layer Flow Past a Constant Wedge within Porous Media

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
Shreenivas R. Kirsur ◽  
Achala L. Nargund ◽  
N. M. Bujurke
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Pressure Gradient ◽  
Adverse Pressure Gradient ◽  
Velocity Profiles ◽  
Similarity Solutions ◽  
Positive Pressure ◽  
Boundary Layer Equations ◽  
Similarity Transformations ◽  
Wide Range

The two-dimensional magnetohydrodynamic flow of a viscous fluid over a constant wedge immersed in a porous medium is studied. The flow is induced by suction/injection and also by the mainstream flow that is assumed to vary in a power-law manner with coordinate distance along the boundary. The governing nonlinear boundary layer equations have been transformed into a third-order nonlinear Falkner-Skan equation through similarity transformations. This equation has been solved analytically for a wide range of parameters involved in the study. Various results for the dimensionless velocity profiles and skin frictions are discussed for the pressure gradient parameter, Hartmann number, permeability parameter, and suction/injection. A far-field asymptotic solution is also obtained which has revealed oscillatory velocity profiles when the flow has an adverse pressure gradient. The results show that, for the positive pressure gradient and mass transfer parameters, the thickness of the boundary layer becomes thin and the flow is directed entirely towards the wedge surface whereas for negative values the solutions have very different characters. Also it is found that MHD effects on the boundary layer are exactly the same as the porous medium in which both reduce the boundary layer thickness.

An Accurate Calculation Method for Two-dimensional Incompressible Laminar Boundary Layers, Including Cases with Regions of Sharp Pressure Gradient

1977 ◽  
Vol 28 (3) ◽  
pp. 149-162 ◽  
Author(s):  
N Curle
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Calculation Method ◽  
Boundary Layer Separation ◽  
Pressure Gradients ◽  
Accurate Calculation ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Displacement Thickness ◽  
Type Boundary

SummaryThe paper develops and extends the calculation method of Stratford, for flows in which a Blasius type boundary layer reacts to a sharp unfavourable pressure gradient. Whereas even the more general of Stratford’s two formulae for predicting the position of boundary-layer separation is based primarily upon an interpolation between only three exact solutions of the boundary layer equations, the present proposals are based upon nine solutions covering a much wider range of conditions. Four of the solutions are for extremely sharp pressure gradients of the type studied by Stratford, and five are for more modest gradients. The method predicts the position of separation extremely accurately for each of these cases.The method may also be used to predict the detailed distributions of skin friction, displacement thickness and momentum thickness, and does so both simply and accurately.

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