Discontinuous solutions of the unsteady boundary-layer equations for a rotating disk of finite radius

In this article, we propose a high order method for solving steady and unsteady two-dimensional laminar boundary-layer equations. This method is convergent of sixth-order of accuracy. It is shown that this method is unconditionally stable. The unsteady separated stagnation point flow, the Falkner–Skan equation and Blasius equation are considered as special cases of these equations. Numerical experiments are given to illustrate our method and its convergence.

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

Journal of Fluid Mechanics ◽

10.1017/s0022112085003470 ◽

1985 ◽

Vol 160 ◽

pp. 257-279 ◽

Cited By ~ 6

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.

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Transition near the edge of a rotating disk

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.578 ◽

2013 ◽

Vol 737 ◽

Cited By ~ 12

Author(s):

Benoît Pier

Keyword(s):

Boundary Layer ◽

Boundary Condition ◽

Large Data ◽

Rotating Disk ◽

Finite Radius ◽

Objective Criterion ◽

Data Set ◽

Wide Range ◽

Stabilizing Effect ◽

Strong Source

AbstractThe rotating-disk boundary layer is generally considered as an example of a flow that displays a robust transition from laminar to turbulent régimes. By taking into account disks of finite radius, Healey (J. Fluid Mech., vol. 663, 2010, pp. 148–159) has predicted a stabilizing effect of the boundary condition, but Imayama et al. (J. Fluid Mech., vol. 716, 2013, pp. 638–657) were unable to confirm this prediction experimentally. Following these contradictory results, the present experimental investigation revisits the rotating-disk boundary layer, without any artificially imposed excitation, and studies in further detail the dynamics prevailing in the region closely surrounding the edge of the disk, as well as the flow beyond the disk. Azimuthal mean velocities and fluctuation amplitudes are recorded with small steps in radial and axial directions for a wide range of disk sizes. An objective criterion is used to define the onset of fluctuations consistently over a large data set. Two distinct mechanisms for the onset of fluctuations are identified. In particular, it is found that the flow over the edge of the disk acts as a strong source of fluctuations. Explanations and suggestions for a possible reconciliation of previous studies are given.

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