Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries

Brøns, Morten; Hartnack, Johan

doi:10.1063/1.869881

Cited by 65 publications

(55 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A particular region of the (S, A) parameter space (namely −1 S < 0 and 0 < A < 3.2) was chosen to construct a control space diagram which exhibited several curves representing flow bifurcations at degenerate critical points. These bifurcations were described and interpreted in the context of [13][14][15] who used methods from nonlinear dynamics to analyse and classify critical points arising in two dimensional, incompressible, viscous flows near to and away from boundaries. The eddy generation process described in Part 1 (where one eddy becomes three) will for convenience be referred to as the 'first phase' of eddy generation.…”

Section: Introductionmentioning

confidence: 99%

Eddy genesis and transformation of Stokes flow in a double-lid-driven cavity. Part 2: Deep cavities

Gürcan

Wilson

Savage

2006

Proceedings of the Institution of Mechanical Engineers, Part C:

View full text Add to dashboard Cite

This paper extends an earlier work [Gürcan et al., Proc. Instn. Mech. Engrs, Part C: J. Mech. Eng. Sci., 2003, 217, 353] on the development of eddies in rectangular cavities driven by two moving lids. The streamfunction describing Stokes flow in such cavities is expressed as a series of Papkovich-Faddle eigenfunctions. The focus here is deep cavities, i.e. those with large height-to-width aspect ratios, where multiple eddies arise. The aspect ratio of the fully developed eddies is found computationally to be 1.38 ± 0.05, which is in close agreement with that obtained from Moffatt's [J. Fluid Mech., 1964, 18, 1] analysis of the decay of a disturbance between infinite stationary parallel plates. Extended control space diagrams for both negative and positive lid speed ratios are presented, and show that the pattern of bifurcation curves seen previously in the single-eddy cavity is repeated at higher aspect ratios, but with a shift in the speed ratio. Several special speed ratios are also identified for which the flow in one or more eddies becomes locally symmetric, resulting in locally symmetric bifurcation curves. By superposing two semi-infinite cavities and using the constant velocity damping factor found by Moffatt, a simple model of a finite multiple-eddy cavity is constructed and used to explain both the repetition of bifurcation patterns and the local symmetries. The speed ratios producing partial symmetry in the cavity are shown to be integer powers of Moffatt's velocity damping factor.

show abstract

Section: Introductionmentioning

confidence: 99%

Eddy genesis and transformation of Stokes flow in a double-lid-driven cavity. Part 2: Deep cavities

Gürcan

Wilson

Savage

2006

Proceedings of the Institution of Mechanical Engineers, Part C:

View full text Add to dashboard Cite

show abstract

“…The bifurcations here are cusp bifurcations, where a center and a saddle merge and disappear. 18 The bifurcation diagram is shown in Fig. 4.…”

Section: ͑55͒mentioning

confidence: 99%

Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions

2006

View full text Add to dashboard Cite

We consider the two-dimensional topology of streamlines near a surface where the Navier slip boundary condition applies. Using transformations to bring the streamfunction in a simple normal form, we obtain bifurcation diagrams of streamline patterns under variation of one or two external parameters. Topologically, these are identical with the ones previously found for no-slip surfaces.We use the theory to analyze the Stokes flow inside a circle, and show how it can be used to predict new bifurcation phenomena.

show abstract

“…A nondegenerate critical point in non-divergent flows is either a center or a saddle, depending on the sign of its Hessian. But a degenerate critical point can be any topologic type, including cusp (Brøns and Hartnack 1999).…”

Section: Equivalent Barotropicmentioning

confidence: 99%

Vertical alignment of stagnation points in pseudo-plane ideal flows

Sun

2017

AIP Advances

View full text Add to dashboard Cite

Recent studies of pseudo-plane ideal flow (PIF) reveal a ubiquitous presence of vortex alignment in both homogeneous and stratified fluids, and in both inertial and rotating reference frames as well. The exact solutions of a steady-state PIF model suggest that stagnation points tend to be vertically aligned and the concentric structure represents a fixed-point phenomenon of the Euler equations. Exception occurs in the rotating frame when a flow holds inertial period and skew center becomes possible. Properties of stagnation points based on Morse theory are obtained, leading to a topological explanation of vertical alignment via pressure Hessian. The study thus uncovers a new aspect of vortex behavior in ideal fluid that requires vortex center to align with the direction of gravity when vortex evolution reaches a laminar end state characterized by steady pseudo-plane velocities. Though the phenomenon arises from the constraint of the Euler equations, under specific conditions the topological theory is applicable to viscous fluid and explains the curvilinear tilting of von Kármán swirling vortex.

show abstract

Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries

Cited by 65 publications

References 23 publications

Eddy genesis and transformation of Stokes flow in a double-lid-driven cavity. Part 2: Deep cavities

Eddy genesis and transformation of Stokes flow in a double-lid-driven cavity. Part 2: Deep cavities

Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions

Vertical alignment of stagnation points in pseudo-plane ideal flows

Contact Info

Product

Resources

About