Invariant solutions of the three-dimensional laminar boundary-layer equations on developing surfaces

1976 ◽  
Vol 10 (4) ◽  
pp. 562-569
Author(s):  
V. S. Kaplan
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Three Dimensional ◽  
Invariant Solutions ◽  
Boundary Layer Equations

Singularities in solutions of the three-dimensional laminar-boundary-layer equations

1985 ◽  
Vol 160 ◽  
pp. 257-279 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Unsteady Boundary Layer ◽  
Boundary Layer Flows ◽  
Boundary Layer Equations ◽  
New Type ◽  
Steady Laminar

The three-dimensional steady laminar-boundary-layer equations have been cast in the appropriate form for semisimilar solutions, and it is shown that in this form they have the same structure as the semisimilar form of the two-dimensional unsteady laminar-boundary-layer equations. This similarity suggests that there may be a new type of singularity in solutions to the three-dimensional equations: a singularity that is the counterpart of the Stewartson singularity in certain solutions to the unsteady boundary-layer equations.A family of simple three-dimensional laminar boundary-layer flows has been devised and numerical solutions for the development of these flows have been obtained in an effort to discover and investigate the new singularity. The numerical results do indeed indicate the existence of such a singularity. A study of the flow approaching the singularity indicates that the singularity is associated with the domain of influence of the flow for given initial (upstream) conditions as is prescribed by the Raetz influence principle.

On the nature of unsteady three-dimensional laminar boundary-layer separation

1978 ◽  
Vol 88 (2) ◽  
pp. 241-258 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Layer Separation ◽  
The Body ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Unsteady Separation ◽  
Similar Solutions

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

The asymptotic form of the laminar boundary-layer mass-transfer rate for large interfacial velocities

1962 ◽  
Vol 12 (3) ◽  
pp. 337-357 ◽  
Author(s):  
Andreas Acrivos
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Laminar Boundary Layer ◽  
Mass Transfer Rate ◽  
Three Dimensional ◽  
Main Stream ◽  
Boundary Layer Equations ◽  
Asymptotic Expressions ◽  
General Systems

The convective diffusion of matter from a stationary object to a moving fluid stream is distinct from pure heat transfer because of the appearance of a finite interfacial velocity at the solid surface. This velocity is related to the rate of mass transfer by a dimensionless groupBin such a way that for −1 <B< 0 the transfer is from the bulk to the surface while for 0 <B< ∞ the transfer is from the surface to the main stream. In this paper, asymptotic solutions to the two-dimensional laminar boundary-layer equations are developed for the caseB[Gt ] 1, and for rather general systems. It is shown that in most instances the asymptotic expressions for the rate of mass transfer become accurate whenB> 3 and that the transition region between the pure heat-transfer analogy (B∼ 0) and theB[Gt ] 1 asymptote may be described by a simple graphical interpolation. These results may easily be extended to three-dimensional surfaces of revolution by the usual co-ordinate transformations of boundary-layer theory.

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