Fig. (1) shows the computational domains, and Table (1). lists the computational parameters. For reducing the computational cost, the domain is limited to a fan-shaped region instead of a full circular disk. The narrower domain (255 ≤ r ≤ 772) and wider domain (130 ≤ r ≤ 779) computations with different sizes in the radial direction are carried out. In both cases the domain sizes in other directions are 0 ≤ θ ≤ 2π/32 and 0 ≤ z ≤ 54. For the azimuthal direction, periodic boundary conditions are used, and because the azimuthal size is 2π/32, fluctuations with an azimuthal wavenumber of 32 and its multiples appear in the flow field. For the wall-normal direction, the grids are distributed according to a starting wall-normal location of z ₀ and a geometric series of a common ratio of a. The grid locations in the wall-normal direction are,

(4)

where the n-th grid in the wall-normal direction is n_z. The values of a and z ₀ are given in (Table 1).

Fig. (1) Computational domain.

Table 1
Computational domains for the narrower domain and the wider domain cases. The parameters are the radial, azimuthal and wall-normal grid numbers N_r, N_θ and N_z, respectively, and the grid resolutions ∆r, ∆θ, z₀ and a.

The von Kármán similarity solution is adopted for the base velocity components. As an initial condition, the perturbation velocities and pressure components are all set to zero. For the boundary conditions, the perturbation components are set as u = 0, v = 0 and dw/dz = p at the upper boundary, which are identical to the values in Appelquist et al. [18E. Appelquist, P. Schlatter, P.H. Alfredsson, and R.J. Lingwood, "On the global nonlinear instability of the rotating-disk flow over a finite domain", J. Fluid Mech., vol. 803, pp. 332-355.
[http://dx.doi.org/10.1017/jfm.2016.506] ]. The perturbation velocities are zero at the upstream and downstream boundaries, which are in the radial direction. The periodic boundary condition is applied in the azimuthal direction. The Neumann condition for the pressure is used at all boundaries except for the upper boundary. Sponge regions are used in order to prevent nonphysical reflections at the upstream and downstream boundaries. The sponge function is given by

(5)

where the λ(Re) is defined as

(6)

(7)

Equations (6) and (7) indicate where the sponge functions are in the narrower domain case and the wider domain case, respectively. Therefore, the actual computational region is between the two sponge regions. In the sponge regions, the perturbation velocities are forced to decrease.

An artificial disturbance is added in the form of a wall-normal suction and blowing from the disk surface. The disturbance is defined as,

(8)

where w_a(T θ) is the azimuthal amplitude, and w_T(T) is the temporal amplitude. w_a(T θ) is given by

(9)

Fig. (2) shows the temporal amplitude of the artificial disturbance. The amplitudes of the continuative disturbance to excite the convectively unstable mode at the beginning of the computation are shown by the circles in Fig. (2). At T = 0.01601, the amplitude reaches 0.95762, and after that, this amplitude is maintained.

(10)

Fig. (2) The temporal amplitude of the artificial disturbance wT(T).

In the wider domain case, the disturbance with the maximum amplitude w_a,max of 10^-5V is added at Re = 253 ~ 257. In the narrower domain case, the maximum amplitude w_a,max is 10^-11V, and is added at Re = 283 ~ 287. The wavenumber of disturbance α is 32 for both cases. The curved line in Fig. (2) is the amplitude of the disturbance to excite the globally

unstable mode. The excitation of the globally unstable mode is carried out only in the narrower domain. The temporal amplitude is given by,

(11)

At T = 0.01871, the temporal amplitude reaches its maximum. The maximum amplitudes w_a,max are 10^-6V, 10^-8V and

10^-10V. For the azimuthal amplitude w_a,max 10^-6V case, the disturbance is introduced at Re = 528 ~ 532. When the maximum amplitudes wa,max are 10^-8V and 10^-10V, the disturbances are introduced at Re = 588 ~ 592. In the excitation of global instability, the wavenumber of disturbance α is 64.

A computation is carried out on the flow field where velocity fluctuations deriving from a convective instability are expected to take place, and compared with the experimental results to verify our computer code.

Fig. (3) shows the spatio-temporal development of velocity fluctuations log(v_rms) in the z = 1.3 plain. Strong wall-normal suction and blowing were carried out at the disk surface of Re ~ 255 in order to continuatively introduce a stationary disturbance with an azimuthal wavenumber of 32 into the flow field, which was expected to trigger convective instability downstream. The amplitude of the fluctuation of the wall-normal velocity was 10⁻⁵ of the azimuthal disk-surface velocity of the location. According to the theoretical study by Malik [9M.R. Malik, "The neutral curve for stationary disturbances in rotating-disk flow", J. Fluid Mech., vol. 164, pp. 275-287.
[http://dx.doi.org/10.1017/S0022112086002550] ], the critical Reynolds number for the stationary cross-flow convective type instability is 285.36. In experimental studies, 28 ~ 32 spiral vortices are observed [20S. Imayama, P.H. Alfredsson, and R.J. Lingwood, "A new way to describe the transition characteristics of a rotating-disk boundary-layer flow", Phys. Fluids, vol. 24, no. 3, p. 031701.
[http://dx.doi.org/10.1063/1.3696020] , 21S. Imayama, P.H. Alfredsson, and R.J. Lingwood, "Experimental study of rotating-disk boundary-layer flow with surface roughness", J. Fluid Mech., vol. 786, pp. 5-28.
[http://dx.doi.org/10.1017/jfm.2015.634] ]. In experiments, because of minute roughness on the disk surfaces, stationary disturbances are always supplied. After T = 2.0 in Fig. (3), the contour lines become vertical, which means the fluctuations have stopped growing and the flow field is in a steady state.

Fig. (3) Spatio-temporal development of log(vrms) in the z = 1.3 plane. A strong artificial disturbance with azimuthal wavenumber of 32 is continuatively supplied at Re ~ 255.

Fig. (4) shows the color map of log|v| in z = 1.3 plane at T = 3.5 under the same condition. A finger-like pattern is clearly visible. It should be noted that two finger-like patterns correspond to one spiral vortex because the absolute value of the azimuthal velocity fluctuation is shown in the figure. So, there is only one spiral vortex in the flow field shown in the figure. The number of the spiral vortices will be 32 for the entire disk. The stripe pattern starts at Re = 255, and its strength increases in the radial direction. After Re > 600, a turbulent region can be observed. The location is not far from the locations observed in experiments.

Fig. (4) Color map of log|v| in the z = 1.3 plane at T = 3.5 when the strong disturbance is continuatively introduced at Re ~ 255.

Fig. (5) shows the RMS value of v_rms of the same condition in the z = 1.3 plane at T = 3.5. The distribution of the RMS values shows that until Re ~ 560, the disturbance grows exponentially. This trend is close to the experimental results by Imayama et al. [20S. Imayama, P.H. Alfredsson, and R.J. Lingwood, "A new way to describe the transition characteristics of a rotating-disk boundary-layer flow", Phys. Fluids, vol. 24, no. 3, p. 031701.
[http://dx.doi.org/10.1063/1.3696020] ], in which 32 spiral vortices appeared “naturally” without any artificial disturbances except for the minute imperfectness of the disk surface. The wall-normal distribution of the RMS values of azimuthal velocity fluctuations at Re = 510 is shown in Fig. (6). The distribution matches very well with the experimental data of Imayama et al. [21S. Imayama, P.H. Alfredsson, and R.J. Lingwood, "Experimental study of rotating-disk boundary-layer flow with surface roughness", J. Fluid Mech., vol. 786, pp. 5-28.
[http://dx.doi.org/10.1017/jfm.2015.634] ].

Fig. (5) RMS values of vrms in the z = 1.3 plane at T = 3.5 when the strong disturbance is continuatively introduced at Re ~ 255. The solid line is our simulation result. The exponential fitting lines from experimental result (Imayama et al. [20S. Imayama, P.H. Alfredsson, and R.J. Lingwood, "A new way to describe the transition characteristics of a rotating-disk boundary-layer flow", Phys. Fluids, vol. 24, no. 3, p. 031701. [http://dx.doi.org/10.1063/1.3696020] ]) are given as vrms ~ exp(αRe), where α is the growth rate. For the dashed line and the dotted line, the values of α are 0.058 and 0.017.

Fig. (6) The RMS value of the azimuthal velocity fluctuation at Re = 510 when the strong disturbance is continuatively introduced at Re ~ 255. The profile is normalized by the maximum value.

Fig. (7) shows the spatial power spectra and the temporal power spectra of the azimuthal velocity at z = 1.3 and Re = 510. In Fig. (7.1), the data for the entire disk, which were constructed by copying the data of the 2π/32 domain at T = 3.5, were used to calculate the spatial power spectra. The dominant wavenumber is 32 and its harmonics components are present. This result is the reflection of 32 spiral vortices governing the flow field. Next, 250 temporal data points that cover T = 2.5 ~ 3.5 were used to calculate the temporal power spectra, which are shown in Figs. (7.2 and 7.3). The frequency resolution in these figures is 1. The frequency indicates the wavenumber per one rotation of the disk. The finding that the wavenumber and the frequency are both 32 indicates that the 32 spiral vortices are stationary with respect to the disk surface, which is consistent with the previously reported experimental results. Both in our computation, in which the flow field was excited by artificial disturbance at an upstream location of Re ~ 255, and the experimental data of Imayama et al. [20S. Imayama, P.H. Alfredsson, and R.J. Lingwood, "A new way to describe the transition characteristics of a rotating-disk boundary-layer flow", Phys. Fluids, vol. 24, no. 3, p. 031701.
[http://dx.doi.org/10.1063/1.3696020] , 21S. Imayama, P.H. Alfredsson, and R.J. Lingwood, "Experimental study of rotating-disk boundary-layer flow with surface roughness", J. Fluid Mech., vol. 786, pp. 5-28.
[http://dx.doi.org/10.1017/jfm.2015.634] ], in which the disturbance was only naturally provided from the disk surface, the velocity fluctuations were stationary with respect to the disk surfaces and their wavenumbers or the number of spiral vortices were similar. It can be judged that the velocity fluctuation in this computation grew by the convective instability as in experiments.

Fig. (7) (1) Spatial power spectra of v fluctuations at z = 1.3, T = 3.5 and Re = 510, when a strong disturbance is continuatively introduced at Re ~ 255. (2) Frequency spectra of the wavenumber 32 component measured for T = 2.5 ~ 3.5. (3) Frequency spectra of the wavenumber 64 component.

By carefully examining Fig. (7.3), it can be seen that the wavenumber 64 component has the frequencies of 49 and 64. The frequency of 64 is just the harmonic of the 32 component, but the frequency of 49 is not. This result suggests that a pattern of 64 spiral vortices that is traveling at a slower angular velocity than the disk surface co-exists in the flow field.

A computation to verify our simulation code was carried out and the results were compared to the existing experimental results. It was found that our simulation code could successfully calculate the flow field, which was governed by the stationary mode of convective instability, and the results were in good agreement with the experimental results.

An additional simulation concerning the stationary convective instability was carried out using a much weaker artificial disturbance to excite the flow. The amplitude of the fluctuation of the wall-normal velocity was reduced from 10⁻⁵ to 10⁻¹¹ of the azimuthal disk-surface velocity of the location. The reason for this reduction was to delay the growth of the velocity fluctuations related to stationary and convective instability, which was necessary in order to investigate the relation of this growth with the velocity fluctuations derived from global instability, which will be carried out later in this paper.

Fig. (8) shows the spatio-temporal development of the RMS value log(v_rms) in the z = 1.3 plane when the weaker artificial disturbance was continuatively introduced at Re ~ 285. Fig. (8.1), which shows the unfiltered data, corresponds to Fig. (3) in the stronger disturbance case. A similar trend is observed between the cases with weaker and stronger disturbance, but the turbulent region does not appear downstream in the weaker disturbance case. The filtered data showing only the wavenumber 32 component is presented in Fig. (8.2). The figure is very close to Fig. (8.1). The RMS values of the wavenumber 64 component are shown in Fig. (8.3). The higher harmonics components are much smaller compared to the wavenumber 32 components.

Fig. (8) Spatio-temporal development of log(vrms) in the z = 1.3 plane. The weaker artificial disturbance is continuatively introduced at Re ~ 285.

Fig. (9) shows the color map of log|v| in the z = 1.3 plane at T = 3.0, which is the flow pattern, when the weaker disturbance is introduced at Re ~ 285. In Figs. (9.1 and 9.2), patterns corresponding to 32 spiral vortices can be observed. In Fig. (9.3), spiral patterns corresponding to a wavenumber of 64 can be found, but they are weaker than the wavenumber 32 component.

Fig. (9) Color maps of log|v| in the z = 1.3 plane at T = 3.0 when the weaker disturbance is continuatively introduced at Re ~ 285.

Fig. (10) shows the temporal power spectra of the azimuthal velocity measured at z = 1.3 and Re = 630, when the weaker disturbance is added at Re ~ 285. The data are from T = 2.0 ~ 3.0. For the wavenumber 32 component, the peak frequency is at 32, showing that the spiral vortices of wavenumber 32 are stationary with respect to the disk surface. For the wavenumber 64 component, the frequency peaks are at 32, 51 and 64. It should be noted that the frequency peak of 51 is a traveling mode and not stationary. The frequency of 51 is very close to that of 49 found in Fig. (7.3), when the stronger disturbance was used.

In order to study the global instability, two computations having different locations for the start of the downstream sponge regions, namely Re = 560 and 670, were conducted. The other simulation conditions were the same for both cases. Fig. (11) shows the spatio-temporal development of the RMS value of azimuthal velocity fluctuations log(v_rms) in the z = 1.3 plane. Fig. (11) shows the RMS value for unfiltered azimuthal velocity fluctuations when the sponge region started at Re = 560. Figs. (11.1, 11.2 and 11.3) shows the RMS values for the cases with unfiltered data, the wavenumber 32 component filtered out and the wavenumber 64 component filtered out, respectively, when the sponge region started at Re = 630. In all cases, the short-duration artificial disturbance for the wavenumber 64 component was introduced at Re ~ 530. When the downstream sponge region started at Re = 560, the artificial disturbance is convected downstream, and disappears from this flow field after T = 6.5. On the other hand, when the downstream sponge region started at Re = 670, initially, the artificial disturbance is convected downstream, but after T = 1.0, the velocity fluctuation distribution stops changing. It is the global instability which allows the velocity fluctuation to sustain itself. This result suggests that the source of vibration for the globally unstable mode exists somewhere between Re = 560 and 670. In Fig. (11.2), the contour lines strongly oscillate upstream. This noisy region derives from the wavenumber 32 component as can be found in Fig. (11.3). Comparing Figs. 11.2 and 11.4), it is clear that the wavenumber 64 component is dominant in this case. Figs. (12.1, 12.2 and 12.3) shows the color map of log|v| in the z = 1.3 plane at T = 4.0. This figure also confirms that the wavenumber 64 component is dominant.

Fig. (10) Frequency spectra of v at z = 1.3 and Re = 630, for T = 2.0 ~ 3.0, when the weaker disturbance is continuatively introduced at Re ~ 285.

Fig. (11) Spatio-temporal development of log(vrms) in the z = 1.3 plane for the globally stable and unstable cases. The short-duration artificial disturbance is introduced at Re ~ 530.

Fig. (12) Color maps of log |v| in the z = 1.3 plane at T = 4.0 when a short-duration disturbance is introduced at Re ~ 530.

Fig. (13) shows the temporal power spectra of the azimuthal velocity at z = 1.3 and Re = 630, using data of T = 3.0 ~ 4.0. The peak frequency of 50 is obviously the dominant frequency in the wavenumber 64 component shown in Fig. (13.2). The same frequency of 50 also appears in the wavenumber 32 component spectrum in Fig. (13.1). This peak corresponds to a traveling mode structure that is moving slower than the rotating disk. It should be noted that the frequency measured for the present structure of the traveling mode, namely 50, is very close to the frequencies measured for the two traveling mode structures which were found in the convective instability-dominated flow cases, namely 49 and 51, that were presented above. This result indicates that all these traveling mode structures were grown by the global instability.

Fig. (13) Frequency spectra of v at z = 1.3 and Re = 630, for T = 3.0 ~ 4.0. A short-duration disturbance is introduced at Re ~ 530.

Fig. (14) shows the RMS values of v_rms at z = 1.3 for the last rotation. The growth rates of the RMS values for the convectively unstable flow fields and the globally unstable flow field are similar upstream of Re < 540. However, a difference in the growth rates of the RMS values between the convectively unstable flow fields and the globally unstable flow field is observed downstream of Re > 550. Interestingly, in the convectively unstable flow field in which

Fig. (14) Root mean square values of vrms at z = 1.3. Triangles represent the convectively unstable case with stronger disturbance, circles represent the convectively unstable case with weaker disturbance, and squares are the globally unstable case.

the weaker continuative disturbance is introduced upstream, the traveling mode for the 64 component, which can be grown by the global instability, could not grow as much as that in the globally unstable flow field.

At Re ~ 590, where the source of the vibration for the global instability exists, the short-duration artificial disturbance for the wavenumber 64 component was introduced into the convectively unstable flow field when the weaker continuative disturbance was introduced upstream. Two amplitudes cases for the short-duration artificial disturbance were tried: a case with weaker disturbance and a case with stronger disturbance. The weaker disturbance had v_rms~ 10⁻¹⁰, which was lower than the v_rms of ~ 10⁻⁶ in the convectively unstable flow field at Re = 590. The stronger disturbance had v_rms ~ 10⁻⁴, which was stronger than the v_rms of ~ 10⁻⁶ in the convectively unstable flow field at Re = 590.

Fig. (15) shows the spatio-temporal development of the RMS value log(v_rms) at z = 1.3 for two disturbance cases mentioned above. For T = -0.5 ~ 0.0, the flow field is identical with the convectively unstable flow field for T = 2.5 ~ 3.0 when the weaker continuative disturbance was introduced upstream. Figs. (15.1, 15.3 and 15.5) shows the short-duration weaker disturbance case. Figs. (15.2, 15.4 and 15.6) shows the short-duration stronger disturbance case. In Figs. (15.1 and 15.2), which shows the unfiltered flow field, the short-duration disturbance is grown by the global instability. The diagrams for the 64 component (Figs. 15.5 and 15.6) clearly show that the global instability is excited. The diagrams for the 32 component (Figs. 15.3 and 15.4) show that the convective instability is continuously sustained. Interestingly, in the case of the short-duration weaker disturbance, although the artificial disturbance was much smaller than the convectively unstable flow field, which was stationary, the traveling global instability mode could be excited.

Fig. (16) shows the temporal power spectra of the azimuthal velocity at z = 1.3 and Re = 630, for the last rotation in both types of disturbance. In Figs. 16.1 and 16.3), the short-duration weaker disturbance was introduced at Re = 590. In Figs. (16.2 and 16.4), the short-duration strong disturbance was introduced at Re = 590. In both disturbance cases, the wavenumber 32 component is the same, with a frequency of 32, and the wavenumber 64 component has frequencies of 52 or 50, which are similar to the frequency of 50 in the globally unstable flow field. Therefore, the 32 component was grown as a stationary mode by the convective instability and the 64 component was grown as a the traveling mode by the global instability, in both disturbance cases.