The Result of Modulated Effect on Couette Flow

Under a modulated effect, the inner cylinder rotates with the Ω1(1+εcosω^'t) period. The stability of the modulated Couette flow is primarily affected by modulated amplitude and modulated frequency. This study analyzed different aspect ratios, upper and lower boundaries to determine how they affect the modulated Couette flow. To compare modulated effect with the non-modulated effect, the change rate of the critical Re1 is defined here; Δ =(Re_c−Re₀)/Re₀ , Δ>0 represents the modulation has stabilized.

The experimental value was obtained using the optical measurement method and flow observation method. The solid line on Fig. 3(a) is the theoretical result. At a low frequency, when Δ < 0, the critical Re is lower than that for a non-modulated effect.

The result demonstrates that the modulated effect has an unstable effect on flow.

Additionally, when the frequency continues to decrease, Δ approached the quasi-static limit )−ε/(1+ε . At a medium-to-high frequency, Δ increases as frequency increases.

Generally, in this frequency range, the modulated effect destabilizes the flow. As frequency decreases, destabilization increases.

The error of result in this experiment and theoretical value are relatively large at a medium frequency (Fig. 3(a)). At a high frequency, Δ approaches but remains slightly lower than 0. At that moment, modulated frequency has slightly destabilized the flow.

Figure 3(b) presents the curve of medium stable critical Re at different modulated amplitudes. When the flow is unstable, the disturbance has three statuses. For any amplitude, disturbance σ_i =0 at a low frequency is synchronous. After the flow becomes unstable, the frequency of the flow and base status are the same. As frequency

increases, disturbance changes to a quasi-periodic state. At that moment, σ_i ≠0 and the ratio of the flow frequency to base status frequency is not a rational number. When the frequency increases to an ultra-high frequency, σ_i is multiple times the base status frequency. At that moment, disturbance returns to synchronous.

Figure 4 shows the relationship between relative variable Δ of the Re and amplitude under different frequencies. The solid line in the graph is the experimental result derived using the numerical method. The dotted line in Fig. 4(a) is the quasi-steady limit −ε/(1+ε)when ω→0. At an extra low frequency, if the amplitude is high, destabilization is also high. If the critical Re approaches Re₀/(1+ when ε) frequency is ω=0.063, amplitude increases from 0 to 1.0, and the variable Δ of the Re1 gradually decreases to around −0.5. Additionally, based on the graph, the experimental result and quasi-static limit −ε/(1+ε) are extremely close. When the modulated frequency is ω =0.628 and ω=6.28(Figs. 4(b) and 4(c), respectively), the relative variable Δ of the Re1 will decrease as modulated amplitude increases.

When the modulated amplitude is high, the relative variable Δ of the Re decreases and the flow becomes increasingly unstable. However, the degree of instability is lower than that at a low frequency.

At a high frequency (Fig. 4(d)), when modulated frequency is ω =62.8, the different modulated amplitude does not have a significant effect on relative variable Δ of the Re. The modulated amplitude does not significantly influence flow stability.

For different radius ratios η, Fig. 5 presents the experimental result obtained by this study and Walsh et al.[25]. The graph includes different radius ratios η, which include η = 0.4833, 0.719, 0.88, and a modulated amplitude of 0.5. Regardless of the radius ratio, the stable critical value increases as frequency increases, and approaches a

stable critical value Re under the non-modulated effect. At a low frequency, the ₀ stable critical value approaches Re₀/(1+ . ε)

This study investigated the influence of several aspect ratios h on the stable critical value. Figure 6 presents numerical results. When aspect ratio h decreases, Δ also decreases and flow destabilization increases. When frequency is high, the rate reduction is obvious.

This experiment chose three different cylinder boundary conditions: fixed on sides, top free bottom fix, and top rotate bottom fix. The aspect ratio is h=24. Figure 7 presents the numerical result. The measured results for the three conditions are similar.

The flow is not affected by the upper and lower boundaries. The possible reasons are that measurement points for the optical measurement method are at the middle of the cylinder and the aspect ratio is sufficiently large. Therefore, the effect caused by boundaries near measurement points is extremely small.

For the non-modulated effect, the experimental result for wave number of the stable critical value is α =3.19. When the flow is modulated, the flow observation method is used to measure the wave number when flow exceeds a stable state, and determine whether the wave number changes under a modulated effect. Figure 8 shows numerical results. The wave number first decreases as the modulated frequency increases. The wave number is not a fixed value and changes for different modulated frequencies. The degree of change to experimental data is not as large as the theoretical value; however, the tendencies are the same.

Figure 9 presents the relationship between critical wave number k and frequency _c for different radius ratios and modulated amplitudes. When the flow transforms disturbance from synchronous to quasi-periodic, a discontinuous point exists at amplitude. The axial wave number increases as modulated frequency increases. When

the frequency exceeds the first discontinuous point, axial wave number suddenly decreases and then increases again. When the axial wave number reaches a maximum value, the critical Re also reaches a maximum value, the decrease as modulated frequency increases. Finally, the axial wave number approaches the value of non-modulated rotation at an extra high frequency.

Figure 10 presents the relationship between the Re1 and frequency when amplitude is (a)ε =1 and (b)ε =2 under different radius ratios η. When radius ratio η is small, the change to the critical Re under medium-to-high frequency becomes steep; at a stable status, the offset is large.

Table 1 Comparison list of results from this study and other scholars under the condition of a non-modulated Couette flow

Study method

Aspect ratio

Radius ratio

Critical value

Wave number

h η _{Re 0 α}

Huang[45] Flow visualization and Optical method

24 0.4833 68.75 3.19

Donnelly[3] Torsion measurement

method 4 0.5 68.28

Donnelly[46] Flow visualization 5 0.5 68.57 3.10

Simon and Donnelly[4]

Torsion measurement

method 5 0.5 68.23

Sparrow et al.[29]

Numerical method Infinite 0.5 68.19 3.16

Present Numerical method Infinite 0.4833 67.94 3.17

Present Numerical method Infinite 0.5 68.186 3.16

Re

Figure 1 Physics mode graph

( 1 ε cos ω ^t )

Re

+

k

π λ θ =

α λ = π

z

Re

( 1 + ε cos ω t )

z

ξ

d

η η

= −

r 1

−

η

=1 r 1

ξ = 1

ξ = − 1

Figure 2 Coordinate graph

Re

(a)

Figure 3 (a) The relationship of relative variable Δ vs modulated frequency is showing under modulated effect ε =1 (b) A chart showing the variation of Re1 on different modulated amplitudes vs frequencies. (η=0.4833)

(b)

Continued.

(a)

Figure 4 Relationship between relative variable Δ of the Re1 and amplitude under different modulated effects.

(a) ω =0.063 (b) ω =0.628 (c) ω =6.28 (d) ω =62.8

(b)

Continued.

(c)

Continued.

(d)

Continued.

Figure 5 Different inner and outer radius ratios, and the relationship between critical Re and frequency.

Figure 6 Influence of aspect ratio (h) on variable Δ of the critical Re (see Huang[45]).

Figure 7 Influence of different upper and lower boundaries on variable Δ of the critical Re (see Huang[45]).

Figure 8 Relationship between wave number of the modulated Couette flow under a stable critical number and modulated frequency;

upper and lower fixed boundaries are ε =0.5and 4833η=0. , respectively.

(a) η =0.9

Figure 9 Relationship between critical wave number and frequency at different modulated amplitudes.

(a) 9η =0. (b) η =0.4833 (c) η=0.2

(b) η =0.4833

Continued.

(c) η =0.20

Continued.

(a) ε =1

Figure 10 Relationship between the relative variable of the critical Re and frequency under different radius ratios.

(a) ε =1 (b) ε =2

(b)

ε

Continued.