Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk

2007

Quart. Appl. Math.

Self Cite

Abstract. In this paper a theoretical study is undertaken to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible Von Karman's boundary layer flow due to a rotating disk. Particular attention is given to the short-wavelength non-linear non-stationary crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. [497][498][499][500][501][502][503][504][505][506][507][508][509][510][511][512][513] for the stationary linear and non-linear modes, it is revealed here that the nonstationary modes with sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck theory. Making use of this approach, which takes into account the non-linear and non-parallel effects, the asymptotic structure of the non-stationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface. As a consequence of the matching of the solutions in adjacent regions it is found that in the linear case the wavenumber and the orientation of the lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [Proc. Roy. Soc. London Ser. A 406 (1986), 93-106] for the stationary modes. The asymptotic theory shows that the non-parallelism has a destabilizing effect. A Landau-type equation for the modulated vortex amplitude with coefficients that are often difficult to get from finite Reynolds number computations has also been obtained from a weakly non-linear analysis in the limit of infinitely large Reynolds numbers. The non-linearity has also been found to be destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective.

Section: Discussionmentioning

confidence: 99%

Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

2007

Quart. Appl. Math.

Self Cite

“…The equations corresponding to azimuthal velocity expansion are omitted because they are similar to (13). In the above equations U ij is defined by U ij = α i rū j + β iv j ; i, j running from 0 to 1.…”

Section: Region IIImentioning

confidence: 99%

“…The ongoing research on the flow due to a rotating-disk indicates that the underlying crossflow vortex instability may appear in two distinct forms, respectively, absolute and convective. The presence of the former was identified first in [3] and [4], and later extensively investigated in [5][6][7][8][9] and [10][11][12][13]. It can be inferred from these studies that the radially growing absolutely unstable disturbances may constitute a route to transition in rotating-disk flow, even though the wall compliance through surface coating may prevent the occurrence of absolute instability, see for instance [14,15].…”

Section: Introductionmentioning

confidence: 99%

Lower Branch Modes of the Compressible Boundary Layer Flow Due to a Rotating‐Disk

2004

Stud Appl Math

Self Cite

In this article, a theoretical study is pursued to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating-disk. Special attention is focused on to the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [1] for the incompressible stationary modes, it is demonstrated here that the compressible modes having sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck approach. Making use of this rational asymptotic technique, which rigorously takes into account the nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface, as in the incompressible Von Karman's flow. As a consequence of matching successive regions in the asymptotic procedure, it is found that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [1] for the incompressible stationary mode and in [2] for the compressible stationary modes. The nonparallel influences are toward destabilizing all the modes, though the wall insulation and heating are relatively stabilizing for the modes in the vicinity of the stationary mode, unlike the wall cooling. and 9600 Garsington Road, Oxford, OX4 2DQ, UK. M. TurkyilmazogluThe asymptotic compressible data obtained at high Reynolds number limit compares fairly well with the numerical results generated directly solving the linearized compressible system with usual parallel flow approximation.

“…The stability investigations of the rotating-disk boundary layer flow have been the subject of many theoretical studies making use of the parallel flow assumption during the linearization of the governing equations. Research of this kind indicates that besides the well-known convective instability, which may manifest itself in the form of viscous lower branch or inviscid upper branch, as studied comprehensively in [3][4][5][6][7][8][9], the absolute instability of the crossflow vortices may also be a route to transition to turbulence, as investigated in [10][11][12][13] and [14][15][16][17]. A more rigorous approach based on a large Reynolds number assumption for the formal asymptotic solutions of the full nonlinear governing equations, and taking care of the nonparallelism of the basic flow was later developed, known as the triple-deck theory; see for instance [18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Effects of Suction on the Nonstationary Lower Branch Modes of a Compressible Boundary Layer Flow Due to a Rotating Disk

2005

Stud Appl Math

Self Cite

In this paper, suction and injection effects are investigated theoretically on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating disk. In a recent study [1], it was demonstrated that the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies can be described by an asymptotic expansion procedure as set up in [2] for the incompressible stationary modes, which rigorously takes into account the nonparallel effects. Employing this rational asymptotic technique, it is shown here that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation that is under the strong influence of a suction/injection parameters, which, when set to zero, the relation turns out to be the one obtained previously by Turkyilmazoglu [1] for zero-suction compressible modes.The boundary layer growth contributes in the way of destabilizing all the modes, in particular for the compressible modes, though the wall cooling in the case of suction and the wall insulation and heating in the case of injection are found to persist to the destabilization for the modes in the vicinity of the stationary mode. From a linear stability analysis point of view, suction is found to be stabilizing, whereas injection enhances the instability as compared to the Address for correspondence: M. Turkyilmazoglu, and 9600 Garsington Road, Oxford, OX4 2DQ, UK. 358M. Turkyilmazoglu no suction through the surface of the disk. In both cases, positive frequency waves are found to be highly destabilized as compared to the waves having negative frequencies. The findings of the work are also fully supported after a comparison between the numerical results obtained from directly solving the linearized compressible system with a usual parallel flow approximation and the asymptotic compressible data obtained at a high Reynolds number.