4.2.1. Convective instability

Figure 7 displays disturbance time histories, at four successive radial locations, for several modulation parameter settings with Reynolds number ${{Re}}=500$ and azimuthal mode number $n=32$. The impulse is again centred at $r_f={{Re}}$, while the envelopes of the azimuthal component of the vorticity at the wall, $\pm |\xi _{\theta,w}|$, are plotted for a fixed value of $\theta$. The modulation amplitude is $\epsilon =0.1$, with frequencies $\varphi =4$ (dashed lines), $\varphi =8$ (dot-dashed) and $\varphi =12$ (dotted). Solid lines depict the corresponding disturbance development in the steady von Kármán flow. For each set of disk modulation settings, the disturbance decays rapidly at $r=r_f$. Disturbances then propagate radially downstream away from the impulse centre. At radial positions $r>r_f$, there is an initial period of quiescence (due to a delay in the disturbance reaching these locations), before the disturbance grows rapidly over a finite time interval and subsequently decays at a relatively slow rate. Strong radial growth is clearly evident when the scales of the respective vertical axes in each plot are accounted for. Such behaviour demonstrates the convectively unstable nature of the disturbance for the given flow conditions. Time-periodic modulation establishes oscillations in the disturbance amplitude about each radial position during the time phase when growth occurs. Moreover, the envelopes for modulation frequencies $\varphi =4$ and $12$ are only marginally different to that obtained for the steady flow. Indeed, the effect of time-periodic modulation for these particular frequencies is found to enhance the disturbance magnitude over parts of the evolution. On the other hand, the modulation frequency $\varphi =8$ is found to suppress the growth and amplitude of the disturbance at each radial location shown. Hence a stabilising effect is established in this instance.

Figure 7.

Temporal evolution of the envelopes of the azimuthal vorticity perturbation field at the disk wall, $\pm |\xi _{\theta,w}|$, for impulsively excited disturbances with $n=32$ and $r_f={{Re}}=500$. The modulation amplitude is $\epsilon =0.1$, with frequencies $\varphi =4$ (dashed lines), $\varphi =8$ (dot-dashed) and $\varphi =12$ (dotted). Steady flow results are represented by the solid lines. Here, (a) $r=r_f$, (b) $r =r_f+25$, (c) $r =r_f+50$, (d) $r =r_f+75$, and $T =2{\rm \pi} {{Re}}$ is the non-dimensional time period for the disk rotation.

The above analysis was repeated for the same flow conditions $({{Re}},n)$ and modulation frequencies $\varphi$, but for the modulation amplitude $\epsilon =0.2$. The resulting time histories of the envelopes $\pm |\xi _{\theta,w}|$ are plotted in figure 8, with the line types the same as those presented in figure 7. The stabilising effect brought about by modulating the disk with a frequency $\varphi =8$ is even greater at this larger modulation amplitude, with significant disturbance damping realised at all four radial locations shown. Additionally, a stabilising effect is now evident for the other two modulation frequencies, especially about larger radial locations. However, the greatest stabilising effect is again found for $\varphi =8$, which is consistent with the optimal stabilising frequency found via Floquet theory (see table 4 of Morgan et al. Reference Morgan, Davies and Thomas2021).

Figure 8.

Same as figure 7, but for $\epsilon =0.2$.

The convectively unstable behaviour of the above disturbances, and the stabilising effect induced by modulating the disk rotation rate, are demonstrated further in the spatiotemporal contour plots of $|\xi _{\theta,w}|$ shown in figure 9. Disturbance evolution is plotted in the $(r,t/T)$-plane for the modulation frequency $\varphi =8$, while the amplitude $\epsilon$ increases from zero in figure 9(a) to $0.2$ in figure 9(c) at $0.1$ step intervals. All solutions have been normalised to have a maximum amplitude of unity at $t/T=0.1$. Additionally, contours are drawn using a natural logarithmic scaling, with blue and dark red contours matched to amplitude levels $\ln |\xi _{\theta,w}|=-1$ and $\ln |\xi _{\theta,w}|=7$, respectively. The leading and trailing edges of the three disturbance wavepackets are identified by the outer blue contours originating from the impulse centre $r=r_f$. In each instance, both edges propagate radially outwards with non-zero velocities; the leading edge travels to the right faster than the equivalent trailing edge. Time-periodic modulation of the disk rotation rate establishes disturbance oscillations that are visualised by wiggles in the contours that become more prominent as the modulation amplitude $\epsilon$ increases. Furthermore, modulation has a marked effect on the radial growth of the disturbance, as evidenced by a reduction in the maximum contour level over the given radial range: $\max (\ln |\xi _{\theta,w}|)=7$ (maroon contour) in figure 9(a) for $\epsilon =0$, and $\max (\ln |\xi _{\theta,w}|)=6$ (red contour) in figure 9(c) for $\epsilon =0.2$.

Figure 9.

Spatiotemporal development of $|\xi _{\theta,w}|$ for impulsively excited disturbances with $n=32$ and $r_f={{Re}}=500$. Contours are drawn using a natural logarithmic scaling and normalised at $t/T=0.1$. The modulation frequency is $\varphi =8$, with amplitudes (a) $\epsilon =0$, (b) $\epsilon =0.1$, and (c) $\epsilon =0.2$. (Contour levels are plotted on the logarithmic interval $[-1,7]$.)

Figures 10(c,d) display the magnitude $\max |\xi _{\theta,w}|$ and radial trajectory $r_{max}$ of the wavepacket maximum associated with the disturbances plotted in figure 9. In addition, results for modulation amplitudes $\epsilon =0.3$ and $\epsilon =0.4$ are included, while figures 10(a,b) and 10(e,f) display the equivalent solutions for the respective modulation frequencies $\varphi =4$ and $\varphi =12$. As before, all disturbances are scaled to unity at approximately $t/T=0.1$. Plots of the disturbance maxima $\max |\xi _{\theta,w}|$ further demonstrate the significant stabilisation brought about by modulating the disk rotation rate. At the end of the time interval shown, the maximum disturbance amplitude is reduced by 2–3 orders of magnitude for all three modulation frequencies when the amplitude is $\epsilon =0.4$. A marginally greater stabilising effect is again realised for the modulation frequency $\varphi =8$. Moreover, the plots of the trajectories $r_{max}$ indicate that the maximum of the disturbance amplitude propagates radially outwards at a marginally faster rate as the modulation amplitude $\epsilon$ increases.

Figure 10.

Development of the maximum value of $|\xi _{\theta,w}|$ for disturbance wavepackets generated by impulsive excitation centred at $r_f={{Re}}=500$ for $n=32$ and variable modulation amplitudes $\epsilon$. (a,c,e) Magnitudes of the maxima, (b,d,f) radial trajectories $r_{max}$ of the amplitude maxima, for modulation frequencies (a,b) $\varphi =4$, (c,d) $\varphi =8$, and (e,f) $\varphi =12$. All disturbances are normalised so that they are equal to unity for $t/T =0.1$.

The above analysis demonstrates clearly that the stabilising effect, brought about by a time-periodic modulation of the disk rotation rate, is not limited to the stationary convective instabilities considered by Morgan et al. (Reference Morgan, Davies and Thomas2021). While the behaviour displayed in figures 7–10 is limited to the impulse response of one set of convectively unstable conditions, comparable behaviour was found for many other Reynolds numbers ${{Re}}$ and azimuthal mode numbers $n$ (that are classified as being convectively unstable by a local linear stability analysis). Additionally, a strong stabilising effect was found for other modulation frequencies $\varphi$, consistent with the Floquet analysis of Morgan et al. (Reference Morgan, Davies and Thomas2021).

4.2.2. Absolute instability

It was shown by Lingwood (Reference Lingwood1995, Reference Lingwood1997) that when the radial homogeneous flow approximation has been implemented, linear disturbances in the steady von Kármán flow first become absolutely unstable for Reynolds number ${{Re}} = 507.3$ and azimuthal mode number $n=68$. Consequently, we examine the effects of time-periodic modulation on the development of absolutely unstable disturbances for this critical azimuthal mode number. Figure 11 displays time histories of three perturbations excited impulsively at $r_f={{Re}}$, for Reynolds number ${{Re}}=525$. Solutions are plotted about the four successive radial locations $r_f-25$, $r_f$, $r_f+25$ and $r_f+50$, while the disk modulation frequency is $\varphi =8$, with amplitudes $\epsilon =0$ (solid lines), $\epsilon =0.1$ (dashed) and $\epsilon =0.2$ (dot-dashed). As before, the envelopes of the azimuthal vorticity perturbation field at the disk wall, $\pm |\xi _{\theta,w}|$, are plotted to illustrate the development and growth of the disturbances. Absolute instability is demonstrated clearly for the steady disk (as expected, given the parameter settings) as the disturbance grows about all four radial locations shown, including the radial position upstream of the impulse centre. Hence strong temporal growth is observed in this instance. On the other hand, modulation of the disk rotation rate has a significant effect on the disturbance development. Disturbances are again characterised by oscillations, whereby the amplitude of the perturbation grows and decays in tandem with the unsteady oscillations of the disk. (This behaviour is less obvious for the case $\epsilon =0.2$ due to a considerable reduction in the disturbance magnitude.) For the modulation amplitude $\epsilon =0.1$, the maximum and minimum amplitudes of the disturbance appear to remain constant throughout the time interval shown. This would suggest that over one full period of disk modulation, $T_m$, the disturbance neither grows nor decays. Thus behaviour equivalent to critical absolute instability is expected. Hence the onset of absolute instability has been raised to a larger Reynolds number, and stabilisation of the rotating disk boundary layer has been achieved. Further stabilisation is established for the modulation amplitude $\epsilon =0.2$, where the disturbance magnitude is found to decrease rapidly at every radial location plotted. Thus strong temporal decay is now observed.

Figure 11.

Temporal evolution of the envelopes of the azimuthal vorticity perturbation field at the disk wall, $\pm |\xi _{\theta,w}|$, for impulsively excited disturbances with $n=68$ and $r_f={{Re}}=525$. The modulation frequency is $\varphi =8$, with amplitudes $\epsilon =0$ (solid lines), $\epsilon =0.1$ (dashed) and $\epsilon =0.2$ (dot-dashed), and (a) $r=r_f-25$, (b) $r =r_f$, (c) $r=r_f+25$, and (d) $r =r_f+50$.

It is worth emphasising that the oscillations observed in the perturbations shown in figures 7, 8 and 11 are brought about by the time-periodic modulation of the disk rotation rate, and consequently are dependent on the phase of the initial perturbation with respect to the periodic motion of the disk. For this investigation, the unsteady base flow (2.10) was established for the disk rotation rate $\varOmega ^{*}$ given in (2.2), where a phase shift of ${\rm \pi} /2$ was introduced to ensure that time zero was matched to the steady von Kármán flow conditions. Although the impulse response and phase of these oscillations are governed by the initial time of the impulsive forcing, qualitatively similar stability characteristics are realised for disturbances excited about other phases of the unsteady base flow, as demonstrated in Appendix B.

The spatiotemporal development of the above three disturbances is plotted in figure 12, with blue and dark red contours matched to amplitude levels $\ln |\xi _{\theta,w}|=-1$ and $\ln |\xi _{\theta,w}|=12$, respectively. Again, disturbances have been normalised to unity at $t/T=0.1$. The leading edge of the three disturbance wavepackets propagates radially downstream with a commensurate velocity. However, in addition to the formation of oscillatory contour levels, modulation of the disk establishes significant variations in the behaviour of the trailing edge, leading to the development of very different types of instability. For the steady disk, the trailing edge propagates radially inwards, as expected for an absolutely unstable disturbance; eventually, the disturbance spreads across the entire radial range of interest. However, for the modulation amplitude $\epsilon =0.1$, the trailing edge propagates with a near-zero velocity and is parallel to the vertical time $t/T$-axis. Thus behaviour matching critical absolute instability is observed, while for $\epsilon =0.2$, the trailing edge propagates to the right and radially downstream, establishing convectively unstable behaviour.

Figure 12.

Spatiotemporal development of $|\xi _{\theta,w}|$ for impulsively excited disturbances with $n=68$ and $r_f={{Re}}=525$. The modulation frequency is $\varphi =8$, with amplitudes (a) $\epsilon =0$, (b) $\epsilon =0.1$, and (c) $\epsilon =0.2$. (Contour levels are plotted on the logarithmic interval $[-1,12]$.)

In addition to the changing character of the trailing edge, figure 12 also illustrates the effect of time-periodic modulation on the disturbance amplitude. Over the radial and time intervals shown, the azimuthal wall vorticity achieves a maximum $\ln |\xi _{\theta,w}|\approx 12$ for $\epsilon =0$, which reduces to $\ln |\xi _{\theta,w}|\approx 7$ for $\epsilon =0.2$. Hence modulation of the disk rotation rate both changes the type of instability from absolute to convective and brings about a significant reduction in the magnitude of the disturbance.

Temporal frequencies and growth rates for the above disturbances may be examined by considering the complex-valued quantity

(4.3)\begin{equation} f=\dfrac{{\rm i}\,{{Re}}}{A}\,\dfrac{\partial A}{\partial t}, \end{equation}

where the respective real and imaginary parts of $f$ represent the temporal frequency and growth rate based on the global time non-dimensionalisation. Once again, $A$ is given by the azimuthal vorticity perturbation field at the disk wall, $\xi _{\theta,w}$. Figure 13 displays the temporal frequencies $f_r$ and growth rates $f_i$ for the above three disturbances. Furthermore, results are presented about the four radial locations considered in figure 11: $r_f-25$ (solid lines), $r_f$ (dashed), $r_f+25$ (dot-dashed) and $r_f+50$ (dotted). For the results plotted for the steady disk, depicted in figures 13(a,b), the temporal frequency $f_r$ and growth rate $f_i$ about each radial position approach the same fixed constant for large time: $f\approx -17.0+0.16\text {i}$. (Results to greater accuracy are tabulated in table 2.) This behaviour is to be expected, as all radial positions are equivalent for the base flow (2.10) based on the radial homogeneous flow approximation. Moreover, as the imaginary part of the temporal frequency $f$ is positive for large time, positive temporal growth and absolute instability develop, which is not too surprising given the results of Lingwood (Reference Lingwood1995) and that presented above in figures 11 and 12.

Figure 13.

Temporal frequencies $f_r$ and temporal growth rates $f_i$ for the disturbances modelled in figures 11 and 12, where $n=68$ and $r_f={{Re}}=525$. The temporal development is plotted for four different radial positions: $r =r_f-25$ (solid lines), $r =r_f$ (dashed), $r = r_f + 25$ (dot-dashed) and $r = r_f + 50$ (dotted), and: (a,b) $\varphi =8$ and $\epsilon =0$, (c,d) $\varphi =8$ and $\epsilon =0.1$, and (e,f) $\varphi =8$ and $\epsilon =0.2$.

Table 2.

Mean temporal frequencies $\bar {f}_r$ and growth rates $\bar {f}_i$ for impulsively excited disturbances with $n=68$ and variable Reynolds number ${{Re}}$ and modulation settings $\varphi$ and $\epsilon$. Mean results are calculated by averaging $f_r$ and $f_i$ over one full period of disk modulation about the radius $r=r_f={{Re}}$.

The temporal frequencies and growth rates computed for the two modulated flows, plotted in figures 13(c,d) and 13(e,f), again demonstrate that computations asymptote towards the same solution, irrespective of the radial position. However, as a consequence of the time-periodic modulation of the disk rotation rate, temporal frequencies $f_r$ and growth rates $f_i$ are now characterised by oscillations. Over the span of each modulation cycle $T_m$, the temporal frequency $f_r$ displays periodic behaviour and oscillates about a value near that found for the steady disk. Moreover, the size of these oscillations increases with increasing modulation amplitude $\epsilon$. Similar fluctuations are observed in the temporal growth rates $f_i$, whereby during each modulation cycle, an interval of temporal decay ($f_i<0$) is followed by a period of temporal growth ($f_i>0$). The maximum and minimum values of the temporal growth rate $f_i$, during each modulation cycle, are respectively enhanced and reduced by increases in the modulation amplitude $\epsilon$. Averaging the temporal frequencies $f$ over one period of disk modulation establishes the mean frequencies $\bar {f}\approx -16.9+0.01\text {i}$ and $\bar {f}\approx -16.7-0.44\text {i}$ for the respective modulation amplitudes $\epsilon =0.1$ and $\epsilon =0.2$. (Mean calculations of $\bar {f}$ to greater accuracy are again tabulated in table 2.) Thus in the instance $\epsilon =0.1$, the mean temporal growth rate $\bar {f}_i$ is near critical conditions for absolute instability, while for $\epsilon =0.2$, strong temporal decay ensues.

The above analysis was extended to modulation frequencies $\varphi =4$ and $12$. Linear disturbances were again excited impulsively for Reynolds number ${{Re}}=525$, with modulation amplitudes $\epsilon =0.1$ and $0.2$. Figure 14 displays the corresponding spatiotemporal disturbance development in the $(r,t/T)$-plane. Contour levels and the radial–temporal domain are equivalent to those presented for the modulation frequency $\varphi =8$ in figure 12, to help to draw comparisons. In both instances, time-periodic modulation again establishes a significant stabilising effect. Behaviour consistent with critical conditions for absolute instability is realised for amplitude $\epsilon =0.1$, while convectively unstable behaviour is achieved for $\epsilon =0.2$. On inspecting the disturbance maximum and behaviour of the respective trailing edges, it is clear that the modulation frequency $\varphi =12$ establishes a stabilising effect greater than that found for $\varphi =4$. Indeed, disturbance development obtained for $\varphi =12$ is commensurate with that brought about by modulating the disk at frequency $\varphi =8$. Additional temporal frequencies $\bar {f}$, obtained by averaging over one modulation cycle $T_m$, are included in table 2 for the above disturbances, alongside computations for several other Reynolds numbers ${{Re}}$.

Figure 14.

Same as figure 12 for (a) $\varphi =4$ and $\epsilon =0.1$, (b) $\varphi =12$ and $\epsilon =0.1$, (c) $\varphi =4$ and $\epsilon =0.2$, (d) $\varphi =12$ and $\epsilon =0.2$.

Further disturbances were simulated numerically for Reynolds number ${{Re}}=550$, with modulation parameter settings $\epsilon =0.1,0.2$ and $\varphi =4,8,12$. Figure 15 displays the corresponding temporal frequencies $f_r$ and growth rates $f_i$ computed about the impulse centre $r_f={{Re}}$, with the modulation frequency $\varphi$ increasing monotonically down the figure. The illustration again demonstrates that time-periodic modulation establishes oscillations in both the temporal frequency and growth rate, with the periodicity governed by the modulation frequency $\varphi$ of the disk rotation rate. Additionally, larger modulation amplitudes $\epsilon$ enhance the size of these oscillations. However, the variation in the temporal frequency $f_r$ is found to decrease as the modulation frequency $\varphi$ increases, whereas fluctuations in the temporal growth rate $f_i$ appear to be less affected by the size of $\varphi$.

Figure 15.

Temporal frequencies $f_r$ and temporal growth rates $f_i$ for impulsively excited disturbances with $n=68$ and $r_f={{Re}}=550$. The temporal development is plotted about the radial position $r =r_f$, with the modulation amplitudes $\epsilon =0$ (solid lines), $\epsilon =0.1$ (dashed) and $\epsilon =0.2$ (dot-dashed), for (a,b) $\varphi =4$, (c,d) $\varphi =8$, and (e,f) $\varphi =12$.

The spatiotemporal development associated with the disturbances excited impulsively for Reynolds number ${{Re}}=550$ and modulation frequency $\varphi =12$ are plotted in figures 16(a,c,e,g). In addition to modulation amplitudes $\epsilon =0,0.1,0.2$, the solution for $\epsilon =0.3$ is included in the illustration, while plots in figures 16(b,d,f,h) depict the equivalent behaviour for Reynolds number ${{Re}}=600$. The size of the modulation amplitude $\epsilon$ increases monotonically down the figure, while contour levels are now plotted on the logarithmic interval $\ln |\xi _{\theta,w}|\in [-1,14]$. The type of instability that develops in each instance is again identified by the direction of the leading and trailing edges of the disturbance wavepacket, i.e. upstream or downstream. Absolute instability is observed clearly for the modulation amplitude $\epsilon =0.1$ at both Reynolds numbers considered (figures 16c,d), as the leading and trailing edges propagate in opposite directions. Behaviour near critical conditions for absolute instability is found for $\epsilon =0.2$ and ${{Re}}=600$, as the trailing edge of the wavepacket is almost parallel to the time $t/T$-axis (figure 16f), while for the largest modulation amplitude, $\epsilon =0.3$, convectively unstable behaviour is demonstrated as both leading and trailing edges propagate radially outwards (figures 16g,h).

Figure 16.

Spatiotemporal development of $|\xi _{\theta,w}|$ for impulsively excited disturbances with $r_f={{Re}}$, $n=68$ and $\varphi =12$, for (a) ${{Re}}=550$ and $\epsilon =0$, (b) ${{Re}}=600$ and $\epsilon =0$, (c) ${{Re}}=550$ and $\epsilon =0.1$, (d) ${{Re}}=600$ and $\epsilon =0.1$, (e) ${{Re}}=550$ and $\epsilon =0.2$, (f) ${{Re}}=600$ and $\epsilon =0.2$, (g) ${{Re}}=550$ and $\epsilon =0.3$, (h) ${{Re}}=600$ and $\epsilon =0.3$. (Contour levels are plotted on the logarithmic interval $[-1,14]$.)

Tracing the changes in the contour shading in figure 16 demonstrates a significant reduction in the disturbance amplitude as the modulation amplitude $\epsilon$ increases. The development of the maximum absolute value of the azimuthal perturbation field at the disk wall, $\max |\xi _{\theta,w}|$, is plotted in figures 17(a,c) for the disturbance wavepackets illustrated in figure 16, in addition to the results obtained for the modulation amplitude $\epsilon =0.4$. On normalising all disturbances to unity at time $t/T=0.1$, a huge reduction in the disturbance amplitude is realised by modulating the disk with amplitude $\epsilon =0.4$. For both Reynolds numbers modelled, $\max |\xi _{\theta,w}|$ is of the order $10^{1}$ at the end of one full rotation of the disk for $\epsilon =0.4$, while $\max |\xi _{\theta,w}|$ is of the order $10^{7}\unicode{x2013}10^{9}$ for the steady von Kármán flow. The corresponding radial trajectories $r_{max}$ of the amplitude maxima are plotted in figures 17(b,d), and indicate that the maximum of the disturbance convects further downstream as the modulation amplitude $\epsilon$ increases.

Figure 17.

Development of the maximum value of $|\xi _{\theta,w}|$ for disturbance wavepackets generated by impulsive excitation centred at $r_f={{Re}}$ for $n=68$ and variable modulation amplitudes $\epsilon$. (a,c) Magnitudes of the maxima; (b,d) radial trajectories of the amplitude maxima. The Reynolds number is ${{Re}}=550$ in (a,b), and ${{Re}}=600$ in (c,d). The modulation frequency is $\varphi =12$, and all disturbances are normalised so that they are equal to unity at $t/T=0.1$.

Figures 18(a,c,e) depict the variation in the mean temporal growth rate $\bar {f}_i$ obtained for azimuthal mode number $n=68$, as a function of Reynolds number ${{Re}}$. Results are plotted for all three modulation frequencies modelled so far, i.e. $\varphi =4$, $8$ and $12$. Mean temporal growth rates were determined for Reynolds numbers ${{Re}}\in [500,700]$ at step intervals $\Delta {{Re}}=25$, with modulation amplitudes $\epsilon \in [0,0.3]$. Solutions further demonstrate the significant stabilising effect achieved via the application of time-periodic modulation to the disk rotation rate; mean temporal growth rates $\bar {f}_i$ decrease as $\epsilon$ increases and the onset of absolute instability is raised to larger Reynolds numbers. Moreover, modulation frequencies $\varphi =8$ and $12$ establish a reduction in the mean temporal growth rate greater than the results obtained for $\varphi =4$.

Figure 18.

Mean temporal growth rates $\bar {f}_i$ as functions of Reynolds number ${{Re}}$, with (a) $(n,\varphi )=(68,4)$, (b) $(n,\varphi )=(100,4)$, (c) $(n,\varphi )=(68,8)$, (d) $(n,\varphi )=(100,8)$, (e) $(n,\varphi )=(68,12)$, and (f) $(n,\varphi )=(100,12)$. The modulation amplitudes are $\epsilon =0$ (solid line), $\epsilon =0.1$ (dashed), $\epsilon =0.2$ (dot-dashed) and $\epsilon =0.3$ (dotted). Mean results are calculated by averaging $f_i$ over one full period of disk modulation at the radius $r=r_f={{Re}}$.

Comparable control of absolutely unstable behaviour was realised for other azimuthal mode numbers $n$. For instance, figures 18(b,d,f) depict the equivalent set of computations for $n=100$, where absolute instability is again delayed to larger Reynolds number. Several mean temporal frequencies $\bar {f}=\bar {f}_r+\text {i}\bar {f}_i$ are given in tables 2 and 3, for the two azimuthal mode numbers $n=68$ and $n=100$, and modulation frequencies $\varphi =4,8,12$.

Table 3.

Same as table 2 for $n=100$.

The mean temporal growth rates $\bar {f}_i$, shown in figure 18, are found to decrease proportionally with the square of the modulation amplitude $\epsilon$. This particular observation is illustrated for the modulation frequency $\varphi =8$ in figure 19(a) for $n=68$ and in figure 19(b) for $n=100$. Mean temporal growth rates are plotted as a function of the modulation amplitude $\epsilon$, for four fixed Reynolds numbers, ${{Re}} = 550$ to ${{Re}}=700$ at step intervals $\Delta {{Re}}=50$, with interpolation implemented to generate the curves of best fit. In each instance, solutions can be approximated by a parabolic function in $\epsilon$ of the form

(4.4)\begin{equation} \bar{f}_i=c_1\left(1-\left(\dfrac{\epsilon}{c_2}\right)^{2}\right), \end{equation}

where $c_1$ is the temporal growth rate at $\epsilon =0$, and $c_2$ is the modulation amplitude associated with zero temporal growth. Thus a greater reduction in the temporal growth rate is realised as the modulation amplitude $\epsilon$ increases to larger values.

Figure 19.

Mean temporal growth rates $\bar {f}_i$ as functions of the modulation amplitude $\epsilon$ with frequency $\varphi =8$, and Reynolds numbers ${{Re}}=550$ (solid line), ${{Re}}=600$ (dashed), ${{Re}}=650$ (dot-dashed) and ${{Re}}=700$ (dotted), with azimuthal mode numbers (a) $n=68$, (b) $n=100$.

Critical Reynolds numbers ${{Re}}_c$ for the onset of absolute instability can be estimated by applying linear interpolation to the mean temporal growth rates plotted in figure 18. Table 4 presents predictions for ${{Re}}_c$, for both azimuthal mode numbers $n=68$ and $n=100$. Calculations indicate that modulation frequencies $\varphi =8$ and $12$ raise the critical Reynolds number by approximately 70–85 units in the instance the modulation amplitude $\epsilon =0.2$, whereas a significantly smaller increase in ${{Re}}_c$ is realised for $\varphi =4$, at approximately 14–35 units when $\epsilon =0.2$.

Table 4.

Predictions for the critical Reynolds number ${{Re}}_c$ for azimuthal mode numbers $n=68$ and $n=100$. Calculations are based on applying linear interpolation to the solutions presented in figure 18.

Finally, modulation frequencies $\varphi =4,8,12$ were chosen in this study to demonstrate the control benefits of disk modulation following the earlier observations of Morgan et al. (Reference Morgan, Davies and Thomas2021). In their Floquet analysis, Morgan and co-workers found that the control of stationary convective instabilities (both type I cross-flow and type II Coriolis) was optimised by modulation frequencies $8 \leq \varphi \leq 12$, with negligible control benefits realised for smaller and larger frequencies (see figure 7 of their paper). While extending the above study to other flow conditions and modulation settings would help to identify the optimal modulation configurations for controlling absolutely unstable behaviour, this was unfeasible due to the computational demands of the numerical scheme; a single numerical simulation, for fixed ${{Re}}$, $n$, $\epsilon$ and $\varphi$, required several computational processing hours. Undertaking a thorough investigation of all possible parameters was impractical, and instead the control benefits of modulating the disk rotation rate were presented for those azimuthal modes numbers that are considered to be the most significant on the steady rotating disk, namely $n=32$ for the cross-flow instability (Gregory et al. Reference Gregory, Stuart and Walker1955; Malik Reference Malik1986), and $n=68$ for absolute instability (Lingwood Reference Lingwood1995, Reference Lingwood1996). Nevertheless, further study was undertaken to ascertain whether the frequency range proposed by Morgan and co-workers was optimal for suppressing the temporal growth of absolutely unstable disturbances.

Figure 20 displays mean temporal growth rates $\bar {f}_i$ as functions of the modulation frequency $\varphi$. Solutions are plotted for the two azimuthal mode numbers $n=68$ and $n=100$, in the instance the Reynolds number ${{Re}}=600$; these particular conditions correspond to strong absolutely unstable disturbances on the steady rotating disk. In addition, solutions are plotted for modulation amplitudes $\epsilon \in [0.1,0.3]$ and modulation frequencies $\varphi \in [2,30]$ at step intervals $\Delta \varphi =2$. Cross markers indicate the results obtained for these particular settings, with interpolation implemented to generate the curves of best fit. For the azimuthal mode number $n=68$, results indicate that the optimal frequency for suppressing the temporal growth rate, and the appearance of absolutely unstable behaviour, is found for $\varphi \approx 10$. For smaller and larger modulation frequencies, the effect on the temporal growth rate quickly diminishes, which is comparable with the behaviour found by Morgan and co-workers for the stationary cross-flow instability. Hence the preferred frequency is within the range of optimum frequencies suggested by Morgan et al. (Reference Morgan, Davies and Thomas2021). However, at the larger azimuthal mode number, solutions indicate that a frequency near $\varphi =14$ gives the optimal control benefits. Moreover, similar numerical simulations carried out for azimuthal mode numbers $n=75$, $85$ and $95$, suggest that control is optimised for frequencies $\varphi \approx 11$, $12$ and $13$, respectively. While results suggest that the preferred modulation frequency increases with the size of the azimuthal mode number, comparable reductions in the mean temporal growth rate $\bar {f}_i$ are still realised for $8\leq \varphi \leq 12$. Furthermore, the larger azimuthal mode numbers have not, to these authors’ knowledge, been observed to play a significant role in the laminar–turbulent transition process on a rotating disk. Hence the frequency range suggested by Morgan and co-workers, $8\leq \varphi \leq 12$, is the preferred modulation configuration for controlling both convective and absolute forms of linear instability.

Figure 20.

Mean temporal growth rates $\bar {f}_i$ as functions of the modulation frequency $\varphi$ with Reynolds number ${{Re}}=600$, and modulation amplitudes $\epsilon =0$ (solid line), $\epsilon =0.1$ (dashed), $\epsilon =0.2$ (dot-dashed) and $\epsilon =0.3$ (dotted), and azimuthal mode numbers (a) $n=68$, (b) $n=100$.