Asymptotic features of viscous flow along a corner

Abstract

Abstract.The asymptotic behavior of the equations governing the viscous flow along a right-angle corner is considered. It is demonstrated that consistent asymptotic series exist for the inner corner layer region. These expansions satisfy the corner layer equations and associated boundary conditions. They exhibit algebraic decay of all the flow properties into the boundary layer away from the corner, and prescribe algebraic decay of the cross flow velocities into the outer potential flow. Of course the streamwise velocity and vorticity are constrained to decay exponentially into the potential flow. The form of this algebraic behavior is required in order to facilitate numerical solution of the corner layer equations. Of particular significance is the use of symmetry as a means of providing a boundary condition, predicting the appearance of logarithmic terms, and specifying the occurrence of arbitrary constants. These constants can only be determined from the complete corner layer solution.1. Introduction. The viscous flow along a corner that is formed by the intersection of two perpendicular flat plates has been studied by several authors and has recently been re-examined by Rubin [1] who discusses the past efforts on this problem in some detail. In Rubin's analysis three distinct regions were discernible ( Fig. 1): an irrotational potential flow where the coordinate gradients are in most generality, of equal order; boundary layers where surface normal derivatives are much larger than either of the derivatives in the plane of the surface, and the corner layer where the x-wise gradient alone is small. Solutions for the various sectors were obtained by the method of matched asymptotic expansions utilizing the small parameter (v/2UqX)1/2 = R~1/z. (U0 is the velocity at x -» -®). As is usually the case, the flow near the leading edge x = 0 cannot be obtained in this manner and only asymptotic similarity solutions were considered.Explicit results were presented for the first-order potential flow and first-and second-order boundary layer flows; the corner layer solution was deferred. The asymptotic behavior of the boundary-and corner-layer solutions for min (77, J")-*® is directly related to the nature of the potential flow at the corner point y2 + z2 = 0. 7) and f are the stretched corner layer coordinates; 77 = yRl/2/2x, f = zR1/2/2x. In this connection it was tacitly assumed that the potential flow is regular at y2 + z2 = 0. However, as will be shown, this assumption is unnecessary; with the inclusion of a singularity at the corner point (first appearing in the second-order potential flow) the *

Injection into boundary layers: solutions beyond the classical form

“…This slight reformulation is equivalent to that found in (for example) Stewartson (1954) for the two-dimensional Falkner-Skan problem or Pal & Rubin (1971) for three-dimensional flow along a corner. The above formulation results in the boundary-region equations in the form:…”

“…where ∇ 2 is the two-dimensional Laplacian in the plane spanned by η and ζ. Whilst perhaps less intuitive than the primitive variable formulation, this system is well known for its application to corner boundary-layer flows, see for example Pal & Rubin (1971). It is advantageous to use this formulation because (2.4a) and (2.4b) can be combined to give expressions for the Laplacian of Φ and Ψ.…”

Wall‐cooling effect on linear stability of compressible streamwise corner‐flow

“…The corner is formed by the intersection of two perpendicular flat plates and serves as generic model for various technical applications. The displacement effect of the opposing wall leads to a secondary cross-flow at the far field and a three-dimensional boundary layer in vicinity of the corner line [4,5]. Compressible corner-flow with cooled walls has first been considered in the corner-layer equations by [6] and [7] extended those results to self-similar solutions without the restriction of P r = 1.…”

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“…As it was not practical in the numerical analysis to apply the asymptotic conditions, strictly valid only for the coordinate f -* =°, for values of f excess of ten or twenty, additional consideration of the asymptotic behavior was required in order to describe the proper algebraic decay. A detailed asymptotic analysis is presented by Pal and Rubin [2], where the existence of consistent asymptotic series exhibiting the necessary algebraic behavior is formally demonstrated. Of particular significance for the numerical analysis is the appearance of arbitrary constants and logarithmic terms in these expansions.…”

“…The analysis contained herein is concerned with a numerical solution of the corner layer equations, taking proper account of the asymptotic formulas discussed in [2] The system of equations for the corner layer region is transformed into four Poisson- like equations for the streamwise velocity, suitably modified cross-plane velocities, and a modified streamwise vorticity. This system includes only differentiated forms of the mass continuity equation and vorticity definition and is similar to that described previously by Pearson [3].…”