Results for transient stability analysis

Transient stability analysis for layering modes of purely kinetic model is presented in this subsection. First, Figure 4-3 illustrates the initial transient growth rate Ω_i in the

(

ν₀,ky

)

plane for four different values of e : (a) 0.2, (b) 0.5, (c) 0.8 and (d) 0.9. The cases of Figure 4-3 (b) and (c) _g are often studied in several studies. We add a very dissipative case of Figure 4-3 (a) and a nearly elastic case of Figure 4-3 (d) in this study. We can observe several features of Figure 4-3: (1) a short wavelength (i.e. large wavenumber) can cause a larger value of Ω_i; (2) the increasing value of ν₀ can lead to a larger value of Ω_i; and (3) the higher e (i.e. nearly elastic) can make the value of _g

Ωi enhanced. According to the above feature (3), we can draw additional contour plot of Ω_i in Figure 4-4 for 0.1≤e_g ≤0.9 and 0.001≤k_y ≤1. The feature (1) expresses similar results to those of the paper (cf. p.286 of the A&N 1997). Both indicate that short wavelengths suffer larger initial transient growth rate Ω_i. Those features (1)–(3) have similar tendency to the asymptotic results.

Three parameters of ν₀, e and _g k can cause the magnitudes of _y ω_r^l and Ω_i to increase. This similar phenomenon can be explained by the mathematical derivation. Since the matrix of

[ ]

M _L is

a non-normality matrix and the inner product of each eigenvector cannot be ensured to equal to one.

Hence, the value of Ω_i can be constructed by various combinations. For certain cases, as the value of ω_r^l is a negative value and the corresponding combinative value also appears a negative value, the combinations of above two values will produce a positive value of Ω_i. Then the more negative value of ω_r^l, the more positive value of Ω_i. Therefore, Ω_i has the same tendency of magnitude as ω_r^l in layering modes.

We present the transient growth function G(t) with time in Figure 4-5 for parameters k_y =1, 8

.

=0

eg and ν₀ =0.55. The reason of choosing those parameters is based on the results of Figure 4-3 and 4-4, where the value of Ω_i can possess a larger value as those parameters ν₀, e and _g

k are larger. We consider that the larger value of y Ω_i may be a potential factor to cause enormous peak of transient growth curve. Figure 4-5 illustrates that the function of G(t) decays to zero with time and the maximum peak of G(t) is achieved in very short-term (t≈0.08301). There are three noticeable features of G(t) can be defined: (1) as the function of G(t) asymptotically decays about to 0.1, this time it is referred to as the decayed time and denotes as t , where the subscript “0.1” _0.1 stands for G

( )

t ≈0.1; (2) the maximum value of G(t) for all over time is denoted as G_max; and (3) the time which corresponding to G_max is denoted as t_max. Then we fix the initial transversal wavenumber at k_y =1, the results of t and _0.1 G_max are presented in the following Table 4-1 for different e and _g ν₀. Table 4-1 shows the results of wide ranges of e (0.2 ~ 0.99) for a given _g value of ν₀. Table 4-1 indicates that the decayed time of t just need a short-term to reach as the _0.1 value of e increases to a very elastic value. Therefore, the larger value of _g ν₀ can lead the time of

1 .

t to be a small value. The value of 0 G_max can be increased with the increasing values of e and _g

ν0 and then a significant value of G_max can be obtained in the nearly elastic case (e_g =0.99).

Therefore, the increasing tendency of the maximum transient growth value G_max is similar to that of Ω_i, both are influenced by the increasing values of ν₀ and e . _g

Table 4-1: Values of G_max and t for different parameters _0.1 e and _g ν₀ (k_y =1) 2

.

=0

eg e_g =0.4 e_g =0.5 e_g =0.6 e_g =0.8 99e_g =0. 51

.

0 =0 ν

1 .

t 0 20.557 14.593 11.853 9.286 4.697 0.768

Gmax 4.794 4.917 5.025 5.206 6.266 44.723

55 .

0 =0 ν

1 .

t 0 15.435 10.969 8.917 6.993 3.552 0.594

Gmax 5.919 6.060 6.180 6.383 7.679 55.826

60 .

0 =0 ν

1 .

t 0 8.834 6.283 5.111 4.011 2.044 0.344

Gmax 10.149 10.383 10.572 10.869 12.990 96.374

In addition to Figure 4-5, Figure 4-6 presents the results of small initial transversal wavenumber 05

.

=0

ky . Figure 4-6 (a) shows the short-term evolution of G(t) for t=0~50 and (b) the long-term evolution for t=0~8000. The results of Figure 4-6 indicate that the temporal evolution of G(t) will reach a higher value of G_max by reducing the value of k . It has different results between G and _y

Ωi, both have inverse tendency by changing the value of k . Figure 4-6 (a) illustrates that the _y oscillatory phenomena may occur in the case of smaller k and the function of G(t) seems to need _y a lot of time to decay. Hence, Figure 4-6 (b) presents the completely temporal evolution of G(t) for the behavior of decay. Comparison of the ratio of t between the results of Figure 4-5 and 4-6 is _0.1 about 25000. Such enormous result can be explained by the asymptotic results. In the case of Figure 4-5, the value of ω_r^l ≈−1.28; the other case of Figure 4-6, the value of ω_r^l ≈−5.78×10⁻⁵.

Therefore, since the exponential decay can approach to nearly zero by an approximation (e⁻⁵ ≈0), that is why the value of t of Figure 4-6 needs about 90000 to asymptotically decay. Besides, _0.1 those oscillatory phenomena of G(t) with time are also presented in Figure 4-6. The oscillation occurs due to our system matrix

[ ]

M _L is a non-normality matrix, thus the function of G(t) in several period may be dominated by the trigonometric function. Because we want to understand the effect of each component of χ^v

( )

t with time, we can apply the initial condition which is derived from G_max. We can obtain the temporal evolution of each component with respect to the norm of

( )

t

χ^v . Before we present next figure, we should introduce the expression as follows

( ) ( ) ( )

²

~ 2

t t t

R χ

ν

ν = v , (4-12)

( ) ( ) ( )

( )

²

2

2 ~

~ t

t v t t u R_KE

χ^v

= + , (4-13)

and

( ) ( ) ( )

²

~ 2

t t t T R_T

χ^v

= , (4-14)

where the symbol of ⋅ is denoted as 2-norm which has been introduced in Eq. (3-6) and we omit the subscript “2” to prevent from confusing with the square, ν^~

( )

t , u~

( )

t , v~

( )

t , and T~

( )

t

are the disturbance component of χ^v

( )

t . The function of R stands for the contribution of density which _ν means the ratio of ν~ t

( )

² to χ^v

( )

t ², the function of R_KE stands for the contribution of kinetic energy which means that the ratio of u^~

( )

t ² + ^~v

( )

t ² to χ^v

( )

t ², and the function of R_T represents the contribution of random fluctuation which means that the ratio of T^~

( )

t ² to χ^v

( )

t ².

Then we can present a temporal evolution of Eq. (4-12)–(4-14) for different parameters and observe which one is the dominated term at given time. We use the same parameters as Figure 4-5 for those of Figure 4-7 and adopt the initial condition which is derived from G_max. Figure 4-7 illustrates: (1) the contribution of density dominates the function of G(t) at initial period and disappears when time proceeds; (2) when t≈0.00392, the contribution of kinetic energy replaces the density to be a dominated term and proceeds for all overtime; and (3) the contribution of random fluctuation is unapparent for initial period, and the curve of R_T(t) is increasing to be obvious till t > 0.01. Then we try to explain the above features by physical opinions. When the disturbance waves start to propagate at initial period, particles tend to vary their arrangement. The linear system will be induced to perturbation by varying a density and then the disturbance amplitude of ^ν~

( )

^t causes the function of R_ν

( )

t to dominate at initial period. With time increasing, the propagation of waves is developed by the kinetic disturbance of velocity, thus the square of disturbance of u~

( )

t and v~

( )

t can provide the function of R_KE

( )

t to dominate for all over time.

Although the function of R_T

( )

t does not have a vantage effect to compare with R_KE

( )

t , R_T

( )

t can also ascend with R_KE

( )

t . Since the disturbance of the granular temperature is associated with velocity very closely. As the disturbance of velocity develops notably, the disturbance of the granular temperature can also be grown. Next, we present Figure 4-8 and 4-9. Both follow the same parameters as those of Figure 4-6 for short-term and long-term, respectively. Figure 4-8 illustrates the similar results to those of Figure 4-7, the function of R_ν

( )

t dominates at initial period and then (t≈0.0746) R_KE

( )

t dominates for all over time. Comparison with the results of Figure 4-7 and 4-8, Figure 4-8 shows that the function of R_KE

( )

t completely dominates the linear system after t >

0.0746 and the contribution of R_T(t) tends to zero. Figure 4-9 illustrates the long-term of Figure 4-8. Hence, the smaller wavenumber (k_y =0.05) can cause that the disturbance contribution of

For given identical parameters, we consider whether the initial condition which causes the value of G_max and the value of Ω_i is the same. Figure 4-10 presents the temporal evolution of each component of χ^v

( )

t from the initial condition of Ω_i for those parameters of k_y =1, e_g =0.8 and 55ν₀ =0. . According to the results of Figure 4-10, the function of each component shows different curves from those of Figure 4-7. We also present Figure 4-11 which has the same parameters as those of Figure 4-8 and the results of Figure 4-11 are inconsistent with those of Figure 4-8. Thus we can conclude that the values of G_max and Ω_i are derived from different initial conditions. Figure 4-10 illustrates that the function of R_ν

( )

t just slightly larger than R_KE

( )

t at initial period and the function of ^R_KE

( )

^t will become a dominated term when time proceeds. In the smaller wavenumber case of Figure 4-11, the function of R_KE

( )

t is larger than R_ν

( )

t at initial period and dominates for all over time. Those are different results of Figures 4-7–4-9, because they are caused by different initial conditions. Then we present a contour plot for Figure 4-12 which collects the value of G_max for 0.501≤ν₀ ≤0.6 and 0.001≤k_y ≤1. As shown, Figure 4-12 illustrates a different result from that of Figure 4-3. Two parameters of ν₀ and e can cause the _g results of Figure 4-3 and 4-12 to have a similar tendency, but the parameter of k will induce _y different results between Figure 4-3 and 4-12. According to those two figures, we can observe that short wavelengths tend to cause larger initial transient growth rate Ω_i . Oppositely, long wavelengths are a major factor to achieve maximum transient growth G_max. This result is consistent with the paper (cf. p.286 of the A&N 1997). Then we can obtain that different initial conditions can lead to different results of Ω_i and G_max. We present Figure 4-13 which collects the time of t_max as the function of G(t) achieving a value of G_max. Figure 4-13 presents that the value of G_max can be faster attained (i.e. t_max is smaller) by increasing those parameters ν₀, e and _g k . This result _y

seems to be different from the result of Ω_i. By increasing the values of ν₀, e and _g k , it can _y lead to more larger value of Ω_i but approach to the smaller value of t_max. Therefore, we assume that the value of G_max can be obtained by the value of Ω_i multiplied t_max, and then the results are presented in Figure 4-14 for e_g =0.8. According to Figure 4-14, we can observe that it has similar tendency to Figure 4-12 (c). Hence, we can conclude that although the value of Ω_i is larger at initial period however the value of t_max is too short to develop G_max well.

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 49-55)