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Review Article

Instabilities of the von Kármán Boundary Layer

[+] Author and Article Information
R. J. Lingwood

Department of Mechanics,
Linné FLOW Centre,
Royal Institute of Technology (KTH),
Stockholm SE-100 44, Sweden
e-mail: lingwood@mech.kth.se
University of Cambridge,
Cambridge CB23 8AQ, UK
e-mail: rjl1001@cam.ac.uk

P. Henrik Alfredsson

Department of Mechanics,
Linné FLOW Centre,
Royal Institute of Technology (KTH),
Stockholm SE-100 44, Sweden
e-mail: phal@mech.kth.se

1Corresponding author.

Manuscript received September 10, 2014; final manuscript received January 9, 2015; published online February 18, 2015. Assoc. Editor: Herman J. H. Clercx.

Appl. Mech. Rev 67(3), 030803 (May 01, 2015) (13 pages) Paper No: AMR-14-1075; doi: 10.1115/1.4029605 History: Received September 10, 2014; Revised January 09, 2015; Online February 18, 2015

Research on the von Kármán boundary layer extends back almost 100 years but remains a topic of active study, which continues to reveal new results; it is only now that fully nonlinear direct numerical simulations (DNS) have been conducted of the flow to compare with theoretical and experimental results. The von Kármán boundary layer, or rotating-disk boundary layer, provides, in some senses, a simple three-dimensional boundary-layer model with which to compare other more complex flow configurations but we will show that in fact the rotating-disk boundary layer itself exhibits a wealth of complex instability behaviors that are not yet fully understood.

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References

Figures

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Fig. 1

A sketch (taken with permission from Ref. [5]) of the similarity-solution velocity profiles (note that these are here shown in the laboratory frame) for the von Kármán boundary-layer flow: U, V, and W in the radial, azimuthal, and axial directions, respectively, where z * is the wall-normal direction, r * is the radial direction, and Ω* is the angular speed of rotation of the disk. Dimensional quantities are denoted by *.

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Fig. 2

A copy with permission of Fig. 3.4 of Ref. [6] showing neutral stability curves (ωi = 0, αi = 0) for convectively instability: (a) ωr = −0.0080, (b) ωr = 0, (c) ωr = 0.024, and (d) ωr = 0.080. Types 1 and 2 are marked and ɛ = tan-1(β/(Rαr)) is in degrees.

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Fig. 3

A copy (with permission of the Royal Society) of Fig. 4 of Ref. [8] showing the china-clay record of instability and transition on a rotating-disk (anticlockwise rotation). The stationary vortices are indicated by the white stripes at intermediate radii and the turbulent region is indicated by the white outer annulus; the higher wall shear stresses in the vortex and turbulent region have caused faster evaporation of the methyl salicylate, which was initially used to render the china clay transparent, thus revealing the white china-clay surface coating.

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Fig. 4

A copy with permission of Fig. 45 of Ref. [43] showing a flow over a swept-wing (angle of attack of −4 deg and chordwise Reynolds number of 2.19 × 106). A naphthalene-trichlorotrifluoroethane spray was used to coat a white sublimating layer over the black wing surface. Regions of high shear cause the naphthalene to sublime faster, which then allows the visualization of the stationary crossflow vortices and the zigzagged transition location. Flow is from left to right.

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Fig. 5

A copy with permission of Fig. 15 of Ref. [61] showing: (a) a contour plot of the wave packet envelope, measured at angular increments relative to the initial excitation of θ = 0 deg, 20 deg, 40 deg, 60 deg, 90 deg, 120 deg, 150 deg, 180 deg, 210 deg, 240 deg, 270 deg, 300 deg, 330 deg, and (b) leading edge (×) and trailing edge (○) of the wavepacket trajectory, taken from (a), and least-squares fit to the experimental data (—). The critical Reynolds number for absolute instability is indicated by a dotted line. Nondimensional time, t, is normalized by the period of the disk rotation, T.

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Fig. 6

A copy with permission of Fig. 17 of Ref. [61] showing a contour plot (at four amplitude levels) of the envelope function for stationary disturbances generated by two roughness elements marked by × and ○. The leading and trailing edges of the stationary “wedges” are estimated by dashed lines.

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Fig. 7

A copy (with kind permission from Springer Science and Business Media) of Fig. 16 of Ref. [72] showing a flow visualization from Ref. [25] with numbers 1–4 added to highlight possible evidence of a low-β nonlinear triad resonance in the transitional region. Rotation in the clockwise direction.

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Fig. 8

A copy with permission of Fig. 6 of Ref. [76] showing the dependence of the steep front on the location of the downstream boundary, X = h, for: (a) h = 37, 45, 60, 80, 100 and (b) h = 35.9, 36, 36.2, 36.5, 37, 37.5, 38, 38.5, 39, 39.5, 40

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Fig. 9

A copy with permission of Fig. 6 of Ref. [77] showing the dependence of the transition onset radius, Rt on the edge location, Re = Redge for different values of wall-normal position, Z = z

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Fig. 12

A copy (with kind permission from Springer Science and Business Media) of Fig. 7 of Ref. [93] showing type-1 vortices co-existing with granular spirals in a spin-up experiment (turntable rotating anticlockwise)

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Fig. 11

An extract with permission of Fig. 4 of Ref. [83] showing: (a) a space–time diagram of (1/2)log Vrms from a nonlinear simulation; and (b) a space–time diagram of (1/2)log E(r) from a linear simulation. Vrms is the root mean square of the total azimuthal velocity at a specific radial position, E is the kinetic energy of the disturbances at a specific radial position, r is the dimensionless radius equivalent to R, and time is shown normalized by the number for complete disk rotations.

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Fig. 10

A copy with permission of Fig. 12(a) of Ref. [5] showing the stationary-vortex azimuthal-velocity distribution at z = 1.3 (disk rotation is anticlockwise). Dotted lines indicate R = 500, 600, 700, respectively, moving outward, and the outer solid circle indicates the edge of the disk at R = 731.

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