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

Abstract

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.

The cross-flow instability of the boundary layer on a rotating cone

“…Hence, decreasing ψ is found to destabilize the flow. Similar comparisons for the rotating disk have been given recently for stationary and non-stationary neutral solutions by Turkyilmazoglu (2006Turkyilmazoglu ( , 2007. Figure 12 shows a comparison between the predicted onset of the two instability modes found in the numerical investigation for stationary vortices and the observed onset of spiral vortices as measured in three experimental investigations: Kreith et al (1962), Kappesser et al (1973) and Kobayashi & Izumi (1983).…”

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

“…In spite of the fact that Figure 2 (a-d) represents specific cases of a suction and injection (and restricted Mach numbers), the general trend for values ofs > 0 ands < 0 is similar to those in Figure 2 (a-d). The trends calculated for zero-suction incompressible modes in [40] are preserved here for large blowing, while an opposite behavior is seen for the large suctioning, in particular for an isothermal wall as the Mach number increases. An immediate conclusion to be drawn from Figure 2 (a-d) is that the non-linear effects are destabilizing for a linearly unstable disturbance for all of the Mach numbers within which the neutral stationary or non-stationary waves exist, since B is positive for both suction and injection.…”

“…Regardless of whether the adiabatic or isothermal wall conditions are considered, there exists a region of positive frequencies (and also negative frequencies for the modes under strong wall cooling with suction) that are always unstable. Moreover, much less initial amplitude disturbance is sufficient than for that of the stationary modes determined in [2] and [1], if the perturbations have positive frequencies, an outcome that is also in line with that of [40]. In addition to this, the appearance of double modes for the lower branch curves having positive frequencies as encountered in the numerical stability solution of non-stationary incompressible flow in [24] is clarified here, whose interval of occurrence is particularly increased in the case of suction but reduced in the presence of wall cooling together with injection.…”

“…However, whenever the sign of A is negative as shown in Figure 3 (a-d), an unstable solution can only occur if a disturbance is away from the neutral location, otherwise the amplitude of the disturbance decreases. Applying the same transformations to the amplitude equation (4.12) as in [2], we reach the solution for the amplitude function of any given disturbance as The result (4.15) is a generalization of the stationary and non-stationary non-linear disturbance amplitude, which is valid for both incompressible and compressible flows under either the influence of suction or injection through the surface of the disk, i.e., the cases that were previously investigated by several authors, such as [36], [2], [12], [1], and [40,45]. It can be easily seen from equation (4.15) that for a ≥ 0 (see also Figure 3 (a-d)), the solutions will grow unboundedly and terminate at a finite value of the radius, leading to an unstable solution.…”

“…A further extension of [36] to include the non-stationary lower branch modes has been recently implemented in [39], in which the compressibility effects on the evolution of linear modes were investigated. Non-linearity has been taken into account in [40], and it was shown that a smaller initial finite amplitude of the disturbance with a positive frequency is sufficient to give rise to an exponential growth of the lower branch non-stationary modes, which would then cause the transition to turbulence in the rotating disk boundary layer flow, of course in the absence of other dominant instability mechanisms. The impacts of applying suction or injection through the disk surface were discussed for the absolute instability in the work of [13] and for the convective instability in the works of [12] and [1], but these were limited to the case of the incompressible Von Karman's boundary layer flow.…”