Compressible Modes of the Rotating‐Disk Boundary‐Layer Flow Leading to Absolute Instability

Abstract

This work is devoted to the clarification of the viscous compressible modes particularly leading to absolute instability of the three-dimensional generalized Von Karman's boundary-layer flow due to a rotating disk. The infinitesimally small perturbations are superimposed onto the basic Von Karman's flow to achieve linearized viscous compressible stability equations. A numerical treatment of these equations is then undertaken to search for the modes causing absolute instability within the principle of Briggs-Bers pinching. Having verified the earlier incompressible and inviscid compressible results of [1][2][3], and also confirming the correct match of the viscous modes onto the inviscid ones in the large Reynolds number limit, the influences of the compressibility on the subject matter are investigated taking into consideration both the wall insulation and heat transfer. Results clearly demonstrate that compressibility, as the Mach number increases, acts in favor of stabilizing the boundary-layer flow, especially in the inviscid limit, as far as the absolute instability is concerned, although wall heating and insulation greatly enhances the viscous absolutely unstable modes (even more dramatic in the case of wall insulation) by lowering down the critical Reynolds number for the onset of instability, unlike the wall cooling.

Influence of suction/blowing on the finite amplitude disturbances having non-stationary modes of a compressible boundary layer flow

“…However, we shall concentrate on a more ideal fluid here and use σ = 0.72 throughout in the subsequent analysis. Despite the fact that the selection of the Prandtl number has an influence on the linear and viscous absolutely unstable modes (see for instance [43]), the use of 0.72 as opposed to 1 is believed to have little quantitative effect on our high Reynolds number convective instability results. Two more points are worthy of consideration here.…”

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

“…The resonance mechanism that results from the coalescence of two eigenmodes having zero group velocity and also initially originating from the distinct wavenumber planes in the instability field is known as the absolute instability. 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.…”

“…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) .…”

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

“…This instability was first explored by [9][10][11] and [12]; see also the recent study of [13]. Making use of the Briggs-Bers criterion and assuming that the flow is parallel, the latter authors were able to show that the flow becomes both inviscidly and viscously absolutely unstable.…”