Non-layering modes for purely kinetic model

4.3.1 Results for asymptotic stability analysis

The asymptotic analysis of non-layering modes at large time will be conducted in this subsection. According to the paper (cf. p.276 of the A&N 1997), which points out those results of asymptotic variation of non-layering modes (cf. p.271 of Schmid & Kytomaa 1994) are incorrect.

Since the paper (Schmid & Kytomaa 1994) used the process of the dominance balance is not correct, we will use the same process of the dominance balance proposed by the A&N (1997) to obtain the asymptotic solutions of non-layering modes at large time. The asymptotic solutions ofν^~,u~, v~

andT~

at t→∞ are shown as follows:

e⁻kt

~~

ν , (4-15)

( )

t e ^kt

u~~ 1 ⁻ , (4-16)

( )

t e ^kt v ~ 1 ⁻

~ , (4-17)

and

( )

^{t e kt}

T~~ 1 ^{2 −}

, (4-18)

where 0

2 ₀

0 0 3 >

= +

μ λ

ν

k A . Those parameters A , ₃ λ₀ and μ₀ are defined in Chapter 2.

Accordingly, we can observe that the asymptotic variations ofν^~,u~ and v~ with time at t→∞ are similar to those of νˆ, uˆ and υˆ (cf. 276 of the A&N 1997), respectively. Although the result of asymptotic variation of T~

Eq. (4-18) is different from that of the paper (cf. 276 of the A&N 1997).

However, all the results for the asymptotic temporal evolution of non-layering modes at t→∞ are similar that unbounded dense granular shear flows are asymptotically stable to arbitrary disturbances of non-zero streamwise wavenumber at large time.

4.3.2 Results for transient stability analysis

We show the initial transient growth rate Ω_i of Figure 4-15 in the

(

k ,x ky

)

plane for four cases of e : (a) 0.2; (b) 0.5; (c) 0.8; (d) 0.9. Figure 4-15 illustrates that the value of _g Ω_i approaches to minimum as the initial streamwise wavenumber k equals to the initial transversal wavenumber _x

k . This result is similar to the papers (cf. p.265 of Schmid & Kytomaa 1994; p.287 of the A&N y

1997). This phenomenon represents that as the initial disturbance wave vector starts with 45°

(

kx=ky

)

, the disturbance can possess the least initial transient growth rate Ω_i. Then we can observe Figure 4-15 which shows that the contour values of Ω_i are symmetric with the line of

y

x k

k = . As shown in Figure 4-15, we find the larger value of e can induce the larger contour _g value of Ω_i by comparing with Figure 4-15 (d) and (a). Next, we present an additional figure which can illustrate the effect of ν₀ on Ω_i. Figure 4-16 displays the contours of Ω_i in the

(

ν₀,ky

)

plane for given value of k_x =0.5. Figure 4-16 is divided into four cases of e which are _g the same as those of Figure 4-15. At given value of k_x =0.1 of Figure 4-17, other parameters are set identical with those of Figure 4-16. According to the results of Figure 4-15, we can observe that the minimum value of Ω_i occurs in k_y =0.5 for Figure 4-16 and k_y =0.1 for Figure 4-17.

Then the contour values of those two figures also present a symmetric tendency with k_y =0.5 and 1

.

=0

ky , respectively. Figure 4-16 and 4-17 illustrate that the value of ν₀ is increased with the enhancing contour values of Ω_i. Therefore, we can compare the results of initial transient growth rate between layering modes and non-layering modes. Both can raise the value of Ω_i by increasing the values of ν₀ and e . In the layering modes, increasing value of initial transversal _g wavenumber k leads to a larger value of _y Ω_i. In the non-layering modes, because the effect of initial streamwise wavenumber k is needed to be considered, the larger value of _x k does not _y ensure that the value of Ω_i can also become large. Then we observe the initial wave structure of both modes. Layering modes possess waves which travel only in y-direction, then the short wavelength (i.e. large wavenumber) represents that the faster development of initial disturbances and the disturbance faster decay. However, non-layering modes possess various directions of initial disturbance wave to travel, where the direction of 45° can cause the minimum value of Ω_i. To more understand these results, we should present the transient growth function in the following.

Figure 4-18 shows the temporal evolution of the transient growth function G(t) for two cases: (a)

=1

kx and k_y =1; (b) k_x =0.1 and k_y =1, which are set as e_g =0.5 and ν₀ =0.55. As shown in Figure 4-18, we compare these two figures (a) and (b). The case of Figure 4-18 (a) has a smaller value of G_max and decays faster. It is consistent with the results of Ω_i, the identical component of initial wave vector can lead to smaller Ω and G . Figure 4-18 (b) applies a smaller value of k

which is reduced by one order of k of Figure 4-18 (a), and then we can observe several features _x by this small value of k : (1) the function of G(t) can persist a longer period to decay; (2) the _x maximum peak of G(t) is larger than that of Figure 4-18 (a); and (3) this maximum value G_max does not appear in first peak of G(t), it is shifted to another peak. We consider the effect of ν₀ and then present Figure 4-19. Figure 4-19 shows the temporal evolution of G(t) for ν₀ =0.6 and all parameters are the same as those of Figure 4-18. The results of Figure 4-19 shows that those of Figure 4-18 (b) possess similar three features which are mentioned in the above. The value of G_max and the period of Figure 4-19 are slightly less than Figure 4-18. Next, we also consider the effect of

e for the function of G(t) and present Figure 4-20 for the case of g e_g =0.8. Figure 4-20 illustrates that multiple peaks are unobvious and the value of G_max appears in the first peak which is not significant as that of Figure 4-18 (b). Figure 4-20 still remains longer period of G(t) for small k . _x Therefore, no matter the layering modes or non-layering modes, when increasing the values of coefficient of restitution e and solids volume fraction _g ν₀, two modes have the same results: (1) the initial transient growth rate Ω_i will be increased; and (2) the maximum transient growth G_max will be risen.

Now, we can fix the value of the initial streamwise wavenumber k and define the supremum of _x

Gmax over all possible initial transversal wavenumber k as the optimal growth. Hence, we define _y the value of G as the supremum of ^opt G_max for the given range of 0.01≤k_y ≤1. The superscript

“opt” denotes the optimal growth, the values of k_y^opt and t stand for the condition of ^opt G=G^opt is achieved. According to the results of Figures 4-18–4-20, a significant transient grow occurs in Figure 4-18 (b). Then we use the same parameters as those of Figure 4-18 (b) (e_g =0.5 and

55 .

0 =0

ν ) to seek the optimal results for k_x =0.05 and k_x =0.01. Figure 4-21 illustrates the

following three features: (1) a significant G_max tends to appear in the smaller value of k ; (2) the _x value of G occurs in the smaller value of ^opt k for the smaller _y k ; and (3) the value of _x G_max

decreases faster with k for smaller _y k . This result is similar to that of Figure 1-3. We also list the _x data of optimal growth for k_x =0.05 and 0.01 in the below Table 4-2. Table 4-2 can clearly present similar three features which are mentioned in the above. Then we can present the transient growth function G(t) for the optimal growth. According to the optimal data of Table 4-2, we can obtain Figure 4-22 which is temporal evolution of optimal growth function G . Figure 4-22 illustrates ^opt the similar features to those of Figure 4-18 (b). When optimal growth is achieved: (1) the smaller value of k can possess a significant value of _x G_max; (2) the time of t_max becomes longer; and (3) a significant value of G_max appears to another peak. Therefore, when the value of k approaches _x to an infinitesimal value, the optimal growth G will have tendencies: (1) ^opt k_y^opt tends to be an infinitesimal value; and (2) G tends to be an enormous value. If ^opt G achieves a significant ^opt value, the linear system will have a great possibility to occur unstable. This result satisfies the algebraic instability which is obtained from Section 4-1. We can obtain the temporal development of each component from the initial condition which is derived from G . Figure 4-23 illustrates ^opt that the function of R_ν

( )

t still dominates at initial period and takes over by the function of R_KE

( )

t when time proceeds. Therefore, the significant transient growth G_max is produced by the contribution of kinetic energy R_KE

( )

t for any combination of wave structures.

Table 4-2: Values of G , ^opt t and ^opt k^opt_y for two different k_x 05

.

=0

kx 01k_x =0.

G opt 72.143 131.282

t opt 14.124 46.573

opt

ky 0.58 0.39

In this paragraph, we present a distribution of solids volume fraction disturbance of G for ^opt 05

.

=0

kx . We calculate the real part of ν′=ν^~

( )

^{t exp}

(

^ikx^x+ⁱ

(

^ky−^kx^t

)

^y

)

and present in Figure 4-24.

Figure 4-24 shows the band-structure of solids volume fraction disturbance, where the color of black and white stands for the maximum and minimum values of solids volume fraction disturbance, respectively. Figure 4-24 (a) illustrates that a wavelength is about 126.13 particle diameters and a wave-front possesses the counterclockwise angle of 175.07° from the x-axis. The above results correspond to a wave vector (0.05 , 0.58) for t=0. Next, Figure 4-24 (b) shows the rotated wave vector (0.05 , -0.126) for t=14.124, then a wavelength is about 135.17 particle diameters and a wave-front possesses the counterclockwise angle of 21.61° from the x-axis. Because wave vectors can be turned by the uniform mean shear flows, the angle of wave-front is rotated clockwise from

° 07 .

175 to 21.61° with time. As time increasing to infinite, the transversal wavenumber is larger than the streamwise wavenumber. Then waves only travel in the negative y-direction and the band-structure will be stretched-out to homogeneous layers.

在文檔中 稠密顆粒材料在無邊界均勻剪力流之穩定分析 (頁 55-60)