New solutions for laminar boundary layers with cross flow

Abstract

In some three-dimensional laminar boundary layer problems a coordinate decomposition reduces the governing equations to a primary nonlinear ordinary differential equation describing the streamwise flow in a semi-infinite domain and a secondary linear equation coupled to the primary solution describing the cross flow of infinite spanwise extent. Five new cross-flow problems of this type are investigated within the confines of laminar boundary layer theory. First, the equation for uniform flow transverse to a planar laminar wall jet is found and solved exactly. Second, two solutions for motion transverse to a uniform shear flow along a flat plate are given. A third problem considered is the transverse motion over a flat plate aligned with the uniform mainstream and advancing toward or receding from the mainstream. In the fourth problem, a family of solutions for transverse uniform streams above and below a planar laminar jet is given in closed form. This solution depends on the momentum J of the planar jet and the velocity ratio κ of the transverse streams. The last problem addresses the motion of transverse uniform streams above and below a planar laminar wake. At leading order the cross flow depends only on the velocity ratio κ and not on the drag D produced by the body forming the wake. The influence of the drag first appears at O(x −1 ln x) in the streamwise coordinate expansion of the cross-flow solution.Mathematics Subject Classification (1991). 35C05, 76D10, 76D25.

Crossflow instability of finite Bödewadt flows: Transients and spiral waves

“…͓URL 10. ͑Color online͒ Snapshots of the vertical velocity w of a synchronous axisymmetric state in ͑a͒ a meridional section and ͑b͒ a horizontal section at z = 0.05␥ at a mean Re= 6 ϫ 10 4 modulated by 5% at frequency f = 1.0, together with ͑c͒ a space-time plot of w at z = 0.05␥ with r ͓0,1͔ ͑vertical axis, with r increasing from top to bottom͒ and t ͓0,10͔ ͑horizontal axis, with t increasing from left to right͒ ͑enhanced online͒.…”

“…Both the von Kárman and the Bödewadt crossflows have the feature that they satisfy no slip at a wall and asymptotically tend to zero far above the wall, but there exist other crossflows that do not exhibit this feature. 10 The controversy with the Bödewadt similarity solution is that it requires radial inflow from infinity to approach similarity in the boundary layer as r → 0. 11 The speculation has been that this boundary layer flow may "break down" at some finite radius and hence Bödewadt's similarity solution may not be achieved in practice.…”

Numerical Analysis of Cu + Al 2

“…Recently, Altaç et al [5] studied the mixed convection flow subjected to upward cross flow. Many other significant cross-flow studies can be obtained in these reported research papers [6][7][8][9][10].…”

Heat transfer on a plate beneath an external uniform shear flow

“…Thus it also arises in a three-dimensional boundary layer flow analysis, as reported by Weidman [6]. In that situation the problem is to determine, for a given streamwise uniform shear flow of strain rate β, possible fully-developed cross flows that may exist.…”

“…belong to the classical topics of fluid mechanics. Due to their wide breath of technical and environmental applications [2][3][4] the general research interest in shear driven flows [5][6][7][8] and their heat transfer characteristics [9,10] still remains a viable area of research.…”