Convective and Absolute Instability in the Incompressible Boundary Layer on a Rotating Disk in the Presence of a Uniform Magnetic Field

Jasmine, H. A.; Gajjar, Jitesh S. B.

doi:10.1007/s10665-005-2732-6

Cited by 21 publications

(32 citation statements)

References 17 publications

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“…Such an instability, which was also observed experimentally by [19], manifests itself as a wave pattern of spiral vortices inclined at an higher angle of about 20 • to the radius vector, but with a much reduced wavenumber than that corresponding to the inviscid instability of [12]. The influence of a magnetic field on the convective/absolute instability mechanism was recently studied in [21].…”

Section: Introductionmentioning

confidence: 67%

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

Türkyılmazoğlu

2007

Z. angew. Math. Phys.

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In this study a theoretical approach is pursued to investigate the effects of suction and blowing on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible von Karman's boundary layer flow induced by a rotating-disk. Particular interest is placed upon the short-wavelength, non-linear and nonstationary crossflow vortex modes developing within the presence of suction/blowing at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [1], the role of suction on the non-linear disturbances of the lower branch described first in [2] for the stationary modes only, is extended in order to obtain an understanding of the behavior of non-stationary perturbations. The analysis using the rational asymptotic technique based on the triple-deck theory enables us to derive initially an eigenrelation which describes the evolution of linear modes. The asymptotic linear modes calculated at high Reynolds number limit are found to be destabilizing as far as the non-parallelism accounted by the approach is concerned, and they compare fairly well with the numerical results generated directly by solving the linearized system with the usual parallel flow approximation. An amplitude equation is derived next to account for the effects of non-linearity. Even though the form of this equation is the same as that of found in [2] for no suction, it is under the strong influence of suction and blowing. This amplitude equation is shown to be adjusted by a balance between viscous and Coriolis forces, and it describes the evolution of not only the stationary but also the non-stationary modes for both suction and injection applied at the disk surface. A close investigation of the amplitude equation shows that the non-linearity is highly 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 at destabilization of the flow under consideration. Finally, a smaller initial amplitude of a disturbance is found to be sufficient for the non-linear amplification of the modes in the case of suction, whereas a larger amplitude is required if injection is active on the surface of the disk.

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Section: Introductionmentioning

confidence: 67%

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

Türkyılmazoğlu

2007

Z. angew. Math. Phys.

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“…The disturbance components of the above system are determined later by solving the form of the Navier-Stokes equations that results from substituting these quantities in (1)(2)(3)(4)(5)(6), and subtracting out the mean flow equations, satisfying (8)(9). Having linearized the equations for small perturbations, we find that the linearized Navier-Stokes operator has coefficients independent of θ , and hence, the disturbances can be decomposed into a normal mode form proportional to e iR(βθ−ωt) .…”

Section: Linear Stability Equationsmentioning

confidence: 99%

“…This mechanism is well-documented in the literature for a non-conducting Von Kármán flow; see for instance [3][4][5] amongst many others. Moreover, the recent work of Jasmine and Gajjar [6] concluded that the presence of a normal magnetic field acts in the way of stabilizing the absolute instability mechanisms.…”

Section: Introductionmentioning

confidence: 99%

“…In the current paper we consider the rotating-disk boundary-layer flow in an electrically conducting fluid, in the presence of a magnetic field applied in the direction normal to the disk surface, similar to the work of Jasmine and Gajjar [6]. To derive a self-similar form for the mean flow, we make use of the assumptions that the effects of the electric field are of no significance, and that the fluid motion can not alter the applied magnetic field, being constant everywhere.…”

Section: Introductionmentioning

confidence: 99%

“…It is further assumed that the magnetic field is strongly influenced by diffusive effects because of a low-magnetic-field Reynolds number. Under these conditions, our main interest lies here in the determination of possible resonance instabilities that are not only restricted to the absolute or convective types as investigated in [6], but include the direct resonance instability mechanisms due to the spatial or temporal development of the modes on which the present study focusses. As discussed above, the absolute instability results of Jasmine and Gajjar [6] may not be the most relevant from a physics point of view, although their availability serves to validate our numerical scheme as well as justify our numerical findings.…”

Section: Introductionmentioning

confidence: 99%

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Resonance instabilities in the boundary-layer flow over a rotating-disk under the influence of a uniform magnetic field

Türkyılmazoğlu

2007

J Eng Math

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Instabilities due to resonating waves in the three-dimensional incompressible viscous/inviscid rotating-disk boundary-layer flow are investigated numerically. The influence of a normal magnetic field on the resonances leading, respectively, to the direct spatial instability, the direct temporal instability and the absolute instability are worked out, in an attempt to determine the physically most significant one. It is found that the magnetic field has a stabilizing influence on all the resonance mechanisms outlined. However, the direct spatial-resonance-instability mechanism persists for low Reynolds numbers, when the flow is still in the laminar regime. Thus, attention should be directed to this specific instability mechanism, which offers a strong route to transition to turbulence by means of triggering the nonlinearity.

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Exact solutions corresponding to the viscous incompressible and conducting fluid flow due to a rotating disk

Türkyılmazoğlu¹

2009

Z Angew Math Mech

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A study is pursued in the present paper for the evaluation of the exact solution of the steady Navier‐Stokes equations governing the incompressible viscous Newtonian electrically conducting fluid flow motion over a disk rotating with a constant angular speed. Instead of the traditional von Kármán's axisymmetric evolution of the flow, the non‐axisymmetric conducting flow is taken into consideration here. The three‐dimensional equations of motion are treated analytically yielding derivation of exact solutions which differ from those of corresponding to the classical von Kármán's conducting flow. The effects of Alfven number on the flow field is better conceived from the exact velocity and induced magnetic field equations obtained. Making use of this solution, analytical formulas for the angular velocity and current density components as well as for the magnetic wall shear stresses are extracted. It is proved from the analytical results that the properly defined thicknesses of the flow decay as the magnetic field strength increases in magnitude, further enhanced by the increasing Reynolds number. Interaction of the resolved flow field with the surrounding temperature is then analyzed via the energy equation. The temperature field is shown to accord with the viscous dissipation and the Joule heating. As a result, exact formulas are obtained for the temperature field which takes different forms depending on whether isothermal wall or adiabatic wall conditions are considered. In accordance with the Fourier's heat law, a constant heat transfer from the disk to the fluid takes place, which is found to be effectively increased as the magnetic field becomes dominant on the fluid flow considered.

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Convective and Absolute Instability in the Incompressible Boundary Layer on a Rotating Disk in the Presence of a Uniform Magnetic Field

Cited by 21 publications

References 17 publications

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

Resonance instabilities in the boundary-layer flow over a rotating-disk under the influence of a uniform magnetic field

Exact solutions corresponding to the viscous incompressible and conducting fluid flow due to a rotating disk

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