
Citation: | Zhongmin Huang, M.N. Smirnova, N.N. Smirnov, Zuojin Zhu. 2024: Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model. Journal of Traffic and Transportation Engineering (English Edition), 11(4): 733-746. DOI: 10.1016/j.jtte.2022.08.005 |
Fluid flow throttling is common in industrial and building services engineering. Similar tunnel throttling of vehicular flow is caused by the abrupt number reduction of roadway lane, as the tunnel has a lower lane number than in the roadway normal segment. To predict the effects of tunnel throttling of annular freeway vehicular flow, a three-lane continuum model is developed. Lane Ⅲ of the tunnel is completely blocked due to the need of tunnel rehabilitation, etc. There exists mandatory net lane-changing rate from lane Ⅲ to lane Ⅱ just upstream of the tunnel entrance, which is described by a model of random number generated through a golden section analysis. The net-changing rate between adjacent lanes is modeled using a lane-changing time expressed explicitly in algebraic form. This paper assumes that the annular freeway has a total length of 100 km, a two-lane tunnel of length 2 km with a speed limit of 80 km/h. The free flow speeds on lanes Ⅰ, Ⅱ and Ⅲ are assumed to be 110, 100 and 90 km/h respectively. Based on the three-lane continuum model, numerical simulations of vehicular flows on the annular freeway with such a tunnel are conducted with a reliable numerical method of 3rd-order accuracy. Numerical results reveal that the vehicular flow has a smaller threshold of traffic jam formation in comparison with the case without tunnel throttling. Vehicle fuel consumption can be estimated by interpolation with time averaged grid traffic speed and an assumed curve of vehicle performance. The vehicle fuel consumption is lane number dependent, distributes with initial density concavely, ranging from 5.56 to 8.00 L. Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.
equals to uc2/{1−sech[Λlln(ρc2l/ρm)]} | |
traffic sound speed on lane , m/s | |
equilibrium speed at saturation point on segment , m/s | |
CFL | Courant number |
vehicle grid fuel consumption on lane , mL | |
vehicle fuel consumption on lane , L | |
length scale, m | |
average length of cars, m | |
total length of ring road, km | |
tunnel length, km | |
traffic pressure on lane , veh·m/s2 | |
mandatory net lane-changing rate from lane Ⅲ to lane Ⅱ, veh/h | |
product of and traffic flow acceleration on lane , veh/s2 | |
traffic speed on lane , m/s | |
equilibrium speed on segment , m/s | |
second critical speed, m/s | |
speed scale, m/s | |
free flow speed on segment , m/s | |
braking distance on segment , m | |
kth ramp intersection | |
position of tunnel entrance, km | |
position of tunnel exit, km | |
equals to , parameter used to define traffic pressure | |
ratio of lane-changing time to lane averaged relaxation time | |
random number used to define the mandatory net lane-changing rate | |
equals to cτl/uc2 | |
lane averaged traffic density, veh/km | |
traffic density on lane , veh/km | |
traffic jam density, veh/km | |
random variable to describe ramp flow | |
relaxation time on segment , s |
Fluid flow throttling is common in industrial and building service engineering. In transportation science and engineering for vehicular flow, it is renamed as bottleneck. A review of mathematical modeling of vehicular flows indicates that many models have been developed, as reported by Zhang et al. (2018, 2023).
After considerations for macroscopic modeling of multi-lane vehicular flow reported by Michalopoulos et al. (1984), Helbing and Greiner (1997) put forward a macroscopic model of multi-lane traffic flow from a Boltzmann-like gas-kinetic approach. Daganzo (2002) developed a macroscopic behavior theory of traffic dynamics for homogeneous multi-lane freeways. Focusing on the onset of congestion, Pestov et al. (2019) examined the performance of multi-lane freeway traffic on entering road crossings. Smirnova et al. (2014) proposed a multi-lane continuum model, in which an expression of the parameter to describe the acceleration component in the perpendicular direction in relation to lane-changing was derived.
For the studies on bottleneck effects, Goñi-Ros et al. (2016) found that to make reduction of congestion at sags, a highly effective and innovative way is to use cooperative adaptive cruise control systems. Jin (2018) developed a behavioral kinematic wave model for better design and control strategies to enhance the performance of tunnel bottlenecks.
Recently, Li et al. (2023) have investigated freeway tunnel effects on ring road traffic flow by proposing a two-lane traffic model. Huang et al. (2023) have explored the effects on vehicular fuel consumption of a freeway work zone which has a length of 100 m and occupies lanes Ⅲ and Ⅳ by a four-lane model.
The aim of this paper is to predict the effects of tunnel throttling of annular freeway vehicular flow by developing a three-lane continuum model, where the net lane-changing rate between adjacent lanes is described by a vehicular lane-changing time model. The mandatory lane-changing rate from lane Ⅲ to Ⅰ just upstream of the tunnel entrance is described be a random number model, with the random number generated in a small range around a median based on golden section analysis. For vehicular flows on an annular freeway with a tunnel having a length of 2 km, but with just two lanes thus inducing the throttling of vehicular flow. Numerical simulations with a reliable numerical method of 3rd-order accuracy are conducted to provide spatiotemporal evolution of traffic density, and grid traffic speed. The grid traffic speed on different lanes is used to calculate the mean travel time and its fluctuation through each lane. With an artificially-assumed vehicle performance curve relating to vehicle fuel consumption on vehicular speed, both the vehicle fuel consumption through each lane of the annular freeway and the additional vehicle fuel consumption in comparison with the case of lane blockage vanished (LBV) (i.e., the absence of lane blockage or without tunnel throttling) are predicted.
Numerical results reveal that tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling. Under the conditions in this paper, the density threshold of traffic jam formation normalized by jam density is around 0.19, while for the case without tunnel throttling the threshold is 0.22. The ramp effect slightly delays the occurrence of traffic jam, making the threshold become 0.20. The mean travel time for the case without tunnel throttling is slightly longer than that for the case with tunnel throttling. As soon as the tunnel has originated a traffic jam at the tunnel entrance, the tunnel mean travel time is approximately equal to 0.062 h, suggesting that the time averaged traffic speed through the tunnel is around 32.26 km/h, which agrees well with existing data, suggesting that the three-lane model has the application potential for analyzing the effect of tunnel throttling of vehicular flow and understanding the tunnel bottleneck phenomenon.
In this paper, model equations are presented in Section 2, the numerical method is described in Section 3, numerical results of vehicular flows on freeway with tunnel throttling are discussed in Section 4, and some conclusions are provided in Section 5.
In the three lanes Ⅰ, Ⅱ and Ⅲ, for lane number
In the three-lane model, traffic densities
(1) |
where
However, for traffic flow in the tunnel, the lane average density is
(2) |
where
Eqs. (1), (2) indicate that there is a mass conservation of vehicular motion between lanes Ⅰ, Ⅱ and Ⅲ. Similarly,
(3) |
where
Adopting
(4) |
It shows that if the absolute value of traffic densities between the two lanes is less than 1 veh/km, drivers have not attempted to carry out lane-changing so that the ratio
Parameter | Value | Parameter | Value |
vf1 (km/h) | 110 | 0.0736 | |
vf2 (km/h) | 100 | 0.0819 | |
vf3 (km/h) | 90 | 0.0909 | |
vf4 (km/h) | 80 | 0.1021 | |
Xbr1 (m) | 73 | 0.6525 | |
Xbr2 (m) | 65 | 0.6374 | |
Xbr3 (m) | 58 | 0.6190 | |
Xbr4 (m) | 51 | 0.5985 | |
cτ1 | 5.147 | 40 | |
cτ2 | 4.879 | 60 | |
cτ3 | 4.582 | 20 | |
cτ4 | 4.280 | 50 | |
τ1 (s) | 7.762 | 52 | |
τ2 (s) | 9.007 | Lt (km) | 2 |
τ3 (s) | 10.657 | lveh (m) | 5.8 |
τ4 (s) | 12.834 | uc2 (km/h) | 18 |
ρm (veh/km) | 172 | Imax | 1001 |
l0 (m) | 100 | γ | 0.382 |
v0 = ρ*2vf2 (m/s) | 2.2756 | Δγ | 0.05 |
t0 (s) | 43.945 | g0 = 1−γ | 0.618 |
L (km) | 100 | ||
Note: |
In comparison with existing modules of lane-changing, the present model is simpler than the behavioral theory reported by Daganzo (2002), which has been used to make predictions for separate groups of lanes while recognizing that the traffic stream is usually composed of aggressive and timid drivers. The simplest version of the theory is found to be qualitatively consistent with experimental observations. However, eight observable parameters are used to specify the simple model.
In the normal road segment excluding the tunnel, lane numbers are
(5) |
In the tunnel, the lane number is limited to
(6) |
For
In particular,
(7) |
As an example, when
Details of calculation of the numerical flux with the WENO5 scheme (Henrick et al., 2005; Jiang and Shu, 1996) and time derivative term with the RK3 (Shu, 1988; Shu and Osher, 1988) are reported by Zhang et al. (2021). The numerical method WENO5-RK3 is explicit. The vector form of the equations of the continuum model and the condition of numerical stability are presented as below.
In the normal road segment, the pressure gradient
(8) |
Using
\begin{aligned} & {R}_{{l}^*}={R}_{{l}}+p_{{lx}}=u_l[ \\ & \left.\quad-\left(\rho_{{l}}-\rho\right) /\left(\tau \beta_*\right)+\sigma {q}_3 \delta_{{l3}} / {l}_0+\left(-\delta_{{l3}} / l_0+\delta_{{l} 2} / l_0\right) q_{\mathrm{t} 1} x_{\mathrm{t} 1 \mathrm{w}}\right] \end{aligned} | (9) |
Instead of
(10) |
where
(11) |
In the tunnel,
(12) |
Taking
(13) |
The WENO5-RK3 satisfies the Courant-Friedrichs-Lewy (CFL) condition, the Courant number is given by Eq. (14).
(14) |
where
The Courant number CFL (Shui, 1998) is fixed at 0.3 to ensure numerical stability. The main reason for choosing WENO5 rather than other Riemann solvers is that essentially non-oscillatory (ENO) reconstruction (Shu and Osher, 1988) is based on adaptive stencils, such that the optimal stencil is chosen. This provides high-order accuracy and essentially non-oscillatory behavior. WENO reconstruction (Shu, 1988) consists of a convex combination of all the candidate stencils, and constitutes an improvement on ENO schemes on many levels, as reported by Johnsen and Colonius (2006).
To predict the tunnel throttling effect of annular freeway vehicular flow, numerical simulations are conducted with parameters shown in Table 1. It can be seen that the free flow speeds on lanes Ⅰ, Ⅱ and Ⅲ are
The third column of Table 1 shows the jam density
Case | |||
LBV | 0.000 | 0.000 | 0.00 |
RF0 | 0.000 | 0.000 | 0.00 |
RF1 | −0.191 | −0.191 | 0.05 |
In particular, it is noted that the average length of cars
In the numerical simulations, in the normal segment excluding the tunnel,
(15) |
While in the tunnel,
(16) |
with
To examine the macroscopic model characterized by the ratio of lane-changing time to relaxation time
As the average vehicle length
Comparing Fig. 4(a) and (b) and Fig. 4(c) and (d), the density threshold of traffic jam formation
For
In the tunnel, the density fraction
Correspondingly, Table 3 shows the data of
49.5 | 0.5474 | 0.2509 | 0.5717 | 0.2140 | 0.5391 | 0.2359 | 0.3301 |
49.6 | 0.5476 | 0.2509 | 0.5792 | 0.2055 | 0.5333 | 0.2432 | 0.3298 |
49.7 | 0.5476 | 0.2512 | 0.5889 | 0.1941 | 0.5238 | 0.2544 | 0.3298 |
49.8 | 0.5466 | 0.2529 | 0.6009 | 0.1800 | 0.5098 | 0.2699 | 0.3298 |
49.9 | 0.5418 | 0.2529 | 0.6319 | 0.1852 | 0.2734 | 0.4889 | 0.3744 |
50.0 | 0.5634 | 0.2281 | 0.5835 | 0.2204 | 1.0000 | 0.0000 | 0.4912 |
50.1 | 0.5520 | 0.2330 | 0.5446 | 0.2362 | 1.0000 | 0.0000 | 0.5034 |
50.2 | 0.5242 | 0.2454 | 0.5126 | 0.2509 | 1.0000 | 0.0000 | 0.5056 |
50.3 | 0.4957 | 0.2595 | 0.4899 | 0.2626 | 1.0000 | 0.0000 | 0.5030 |
50.4 | 0.4809 | 0.2675 | 0.4789 | 0.2679 | 1.0000 | 0.0000 | 0.5010 |
50.5 | 0.3846 | 0.3344 | 0.3731 | 0.3448 | 1.0000 | 0.0000 | 0.5076 |
50.6 | 0.3717 | 0.3459 | 0.3722 | 0.3456 | 1.0000 | 0.0000 | 0.4997 |
50.7 | 0.3704 | 0.3471 | 0.3714 | 0.3463 | 1.0000 | 0.0000 | 0.4993 |
50.8 | 0.3691 | 0.3483 | 0.3697 | 0.3478 | 1.0000 | 0.0000 | 0.4996 |
50.9 | 0.3676 | 0.3497 | 0.3675 | 0.3498 | 1.0000 | 0.0000 | 0.5001 |
51.0 | 0.3655 | 0.3517 | 0.3651 | 0.3520 | 1.0000 | 0.0000 | 0.5002 |
51.1 | 0.3632 | 0.3538 | 0.3628 | 0.3541 | 1.0000 | 0.0000 | 0.5003 |
51.2 | 0.3610 | 0.3559 | 0.3606 | 0.3562 | 1.0000 | 0.0000 | 0.5002 |
51.3 | 0.3588 | 0.3579 | 0.3585 | 0.3582 | 1.0000 | 0.0000 | 0.5002 |
51.4 | 0.3567 | 0.3599 | 0.3565 | 0.3601 | 1.0000 | 0.0000 | 0.5002 |
51.5 | 0.3547 | 0.3618 | 0.3544 | 0.3621 | 1.0000 | 0.0000 | 0.5002 |
51.6 | 0.3527 | 0.3638 | 0.3523 | 0.3641 | 1.0000 | 0.0000 | 0.5003 |
51.7 | 0.3507 | 0.3657 | 0.3502 | 0.3662 | 1.0000 | 0.0000 | 0.5004 |
51.8 | 0.3485 | 0.3679 | 0.3479 | 0.3685 | 1.0000 | 0.0000 | 0.5004 |
51.9 | 0.3463 | 0.3700 | 0.3456 | 0.3706 | 1.0000 | 0.0000 | 0.5005 |
52.0 | 0.3438 | 0.3725 | 0.3431 | 0.3732 | 1.0000 | 0.0000 | 0.5005 |
52.1 | 0.2350 | 0.4611 | 0.2375 | 0.4549 | 0.0684 | 0.5235 | 0.4345 |
52.2 | 0.1773 | 0.5494 | 0.1806 | 0.5280 | 0.1214 | 0.5781 | 0.3699 |
52.3 | 0.1584 | 0.6095 | 0.1587 | 0.5860 | 0.1413 | 0.5775 | 0.3456 |
52.4 | 0.1368 | 0.6724 | 0.1362 | 0.6461 | 0.1278 | 0.6388 | 0.3413 |
52.5 | 0.1196 | 0.7320 | 0.1204 | 0.7006 | 0.1147 | 0.6768 | 0.3372 |
52.6 | 0.1108 | 0.7769 | 0.1140 | 0.7361 | 0.1077 | 0.7115 | 0.3333 |
Note: ul (l = 1, 2, 3) is normalized by vf2. |
Downstream of the tunnel end
In this study, travel time is predicted using grid traffic speed as previously reported by Zhang et al. (2018), which is different from the work of Chang and Mahmassani (1988) and Wang et al. (2016). It is noted that in the tunnel lane Ⅲ
In this paper, it is found that in the case of LBV (or the case of three lane tunnel), as shown by the coarse solid purple curve in Fig. 6(a), the mean travel time through lane Ⅱ
0.070 | 0.924 | 1.010 | 1.120 | 0.024 | 0.025 | 0.025 | 0.003 | 0.002 | 0.002 |
0.150 | 1.233 | 1.305 | 1.372 | 0.044 | 0.045 | 0.044 | 0.022 | 0.030 | 0.022 |
0.190 | 1.399 | 1.482 | 1.550 | 0.060 | 0.060 | 0.060 | 0.034 | 0.044 | 0.037 |
0.200 | 1.439 | 1.524 | 1.593 | 0.061 | 0.061 | 0.061 | 0.038 | 0.047 | 0.041 |
0.210 | 1.484 | 1.574 | 1.641 | 0.062 | 0.062 | 0.062 | 0.041 | 0.050 | 0.045 |
0.368 | 2.344 | 2.496 | 2.595 | 0.063 | 0.062 | 0.062 | 0.030 | 0.036 | 0.034 |
0.370 | 2.354 | 2.507 | 2.607 | 0.063 | 0.062 | 0.062 | 0.031 | 0.038 | 0.035 |
In Fig. 6(b), the tunnel mean travel time through lane Ⅱ
As shown in the 6th column of Table 4, the platform height is about 0.062 h, implying that as soon as a traffic jam appears and propagates in the upstream direction, the time-averaged traffic speed through the tunnel
In Fig. 7, the mean travel time
For the case of RF0,
As the free flow speed
In Fig. 7(b)–(d), it can be seen that the coarse solid purple curve denoting the corresponding variable for
When traffic jam originated by the tunnel appears at the tunnel entrance, there is a rarefaction wave propagating in the downstream direction, which decreases the responsiveness of grid traffic speed in the tunnel to the variation of initial density
To predict vehicle fuel consumption by interpolation, time averaged grid traffic speed is used with a vehicular performance curve as shown in Fig. 8. The performance curves for lane Ⅰ, Ⅱ or Ⅲ have steps, reflecting the dependence of vehicle fuel consumption fc on vehicular speed
According to the three vehicle performance curves, with the time averaged grid traffic speed
For
The right-hand side of Fig. 9 indicates that there are corresponding peaks for the troughs of
Taking the sum of fcl(xi), the total vehicle fuel consumption through the annular freeway can be obtained as Eq. (17).
\mathrm{Fu}_l=10^{-3} \sum\limits_{i=1}^{I_{\max }} \mathrm{fc}_l\left({x}_i\right) | (17) |
where the factor
Being consistent with the data given in Table 5, the vehicle fuel consumption through the freeway ring
LBV | RF0 | RF1 | |||||||||
0.070 | 5.965 | 6.970 | 7.959 | 5.974 | 6.987 | 7.968 | 5.975 | 6.987 | 7.968 | ||
0.080 | 5.911 | 6.908 | 7.930 | 5.937 | 6.950 | 7.952 | 5.938 | 6.952 | 7.953 | ||
0.090 | 5.848 | 6.829 | 7.897 | 5.895 | 6.880 | 7.937 | 5.895 | 6.881 | 7.938 | ||
0.190 | 5.572 | 6.397 | 7.189 | 5.656 | 6.505 | 7.316 | 5.644 | 6.490 | 7.306 | ||
0.200 | 5.557 | 6.392 | 7.138 | 5.642 | 6.499 | 7.274 | 5.633 | 6.484 | 7.265 | ||
0.210 | 5.567 | 6.401 | 7.106 | 5.638 | 6.488 | 7.236 | 5.627 | 6.480 | 7.230 | ||
0.260 | 5.724 | 6.410 | 7.061 | 5.765 | 6.471 | 7.166 | 5.746 | 6.460 | 7.152 | ||
0.270 | 6.209 | 6.420 | 7.062 | 6.181 | 6.479 | 7.160 | 6.133 | 6.468 | 7.145 | ||
0.280 | 6.264 | 6.439 | 7.070 | 6.214 | 6.491 | 7.163 | 6.168 | 6.479 | 7.148 | ||
0.300 | 6.327 | 6.481 | 7.086 | 6.264 | 6.521 | 7.166 | 6.246 | 6.507 | 7.156 | ||
0.330 | 6.433 | 6.573 | 7.153 | 6.377 | 6.613 | 7.228 | 6.385 | 6.588 | 7.206 | ||
0.360 | 6.560 | 7.067 | 7.260 | 6.500 | 7.043 | 7.326 | 6.490 | 7.001 | 7.280 | ||
0.368 | 6.596 | 7.098 | 7.291 | 6.535 | 7.081 | 7.354 | 6.516 | 7.027 | 7.291 | ||
0.370 | 6.607 | 7.106 | 7.292 | 6.531 | 7.081 | 7.347 | 6.510 | 7.030 | 7.283 |
Comparing the fuel consumption illustrated by circles and squares for the cases RF0 and RF1, it can be seen that ramp flow merely results in an observable small deviation of
To show the tunnel throttling effect on fuel consumption through the annular freeway more clearly, the difference of fuel consumption to that consumed in the case of LBV called as the additional fuel consumption is calculated by Eq. (18).
(18) |
As shown in Fig. 10(b), the difference
Examining the distributions illustrated in Fig. 10(b), in the cases of RF0 and RF1, if
A three-lane continuum model is developed to predict the effect of tunnel throttling on annular freeway vehicular flow. Based on the continuum model, numerical simulations with a reliable method of 3rd-order accuracy are conducted, with the following conclusions.
(1) The three-lane continuum model is characterized by the vehicular lane-changing time model to describe net lane-changing rate between adjacent lanes, the model of random number generated in a small range around a median from a golden section analysis, is proposed to describe the mandatory lane-changing rate just upstream of the tunnel having two lanes. The three-lane model has the application potential for analyzing the effect of tunnel throttling of vehicular flow and understanding the tunnel bottleneck phenomenon.
(2) Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling. Under the conditions adopted in this study, the density threshold of traffic jam formation in unit of jam density is around 0.19, while for the case without tunnel throttling the threshold is 0.22. The ramp effect slightly delays the occurrence of traffic jam, making the threshold become 0.20.
(3) It is found the mean travel time for the case without tunnel throttling is slightly longer than that for the case with tunnel throttling. If the tunnel has originated a traffic jam at the tunnel entrance, the tunnel mean travel time is approximately equal to 0.062 h, suggesting that the time averaged traffic speed through the tunnel is around 32.26 m/h, which agrees well with existing data.
(4) The time averaged traffic speed can be used to estimate travel time and vehicle fuel consumption through every lane. The vehicle fuel consumption through each lane is lane dependent, can be estimated by interpolation with vehicle performance curve. The vehicle fuel consumption varies concavely with initial density. Under the conditions of in this paper, the vehicle fuel consumption is in the range of 5.56–8.00 L. As a counterpart of comparison, for the case of lane blockage vanished without ramp flow effect, the additional fuel consumption for vehicles through lane Ⅲ is above 0.05 L, when initial density is in the range from 0.10 to 0.36; but for vehicles through lane Ⅱ, the additional fuel consumption is above 0.05 L only when initial density is in the range of 0.09–0.28.
This study is supported by the project of National Natural Science Foundation of China "exploring the road condition effect of travel time using emergency mitigation traffic flow models" (grant 11972341) and fundamental research project of Lomonosov Moscow State University "mathematical models for multi-phase media and wave processes in natural, technical and social systems". We thank Prof. X.Y. Yin and Prof. C.K. Chan respectively at USTC and The Hong Kong Polytechnic University and Dr. Yinglin Li at Peking University for some useful private communications.
Conflict of interest
The authors do not have any conflict of interest with other entities or researchers.
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Parameter | Value | Parameter | Value |
vf1 (km/h) | 110 | 0.0736 | |
vf2 (km/h) | 100 | 0.0819 | |
vf3 (km/h) | 90 | 0.0909 | |
vf4 (km/h) | 80 | 0.1021 | |
Xbr1 (m) | 73 | 0.6525 | |
Xbr2 (m) | 65 | 0.6374 | |
Xbr3 (m) | 58 | 0.6190 | |
Xbr4 (m) | 51 | 0.5985 | |
cτ1 | 5.147 | 40 | |
cτ2 | 4.879 | 60 | |
cτ3 | 4.582 | 20 | |
cτ4 | 4.280 | 50 | |
τ1 (s) | 7.762 | 52 | |
τ2 (s) | 9.007 | Lt (km) | 2 |
τ3 (s) | 10.657 | lveh (m) | 5.8 |
τ4 (s) | 12.834 | uc2 (km/h) | 18 |
ρm (veh/km) | 172 | Imax | 1001 |
l0 (m) | 100 | γ | 0.382 |
v0 = ρ*2vf2 (m/s) | 2.2756 | Δγ | 0.05 |
t0 (s) | 43.945 | g0 = 1−γ | 0.618 |
L (km) | 100 | ||
Note: |
Case | |||
LBV | 0.000 | 0.000 | 0.00 |
RF0 | 0.000 | 0.000 | 0.00 |
RF1 | −0.191 | −0.191 | 0.05 |
49.5 | 0.5474 | 0.2509 | 0.5717 | 0.2140 | 0.5391 | 0.2359 | 0.3301 |
49.6 | 0.5476 | 0.2509 | 0.5792 | 0.2055 | 0.5333 | 0.2432 | 0.3298 |
49.7 | 0.5476 | 0.2512 | 0.5889 | 0.1941 | 0.5238 | 0.2544 | 0.3298 |
49.8 | 0.5466 | 0.2529 | 0.6009 | 0.1800 | 0.5098 | 0.2699 | 0.3298 |
49.9 | 0.5418 | 0.2529 | 0.6319 | 0.1852 | 0.2734 | 0.4889 | 0.3744 |
50.0 | 0.5634 | 0.2281 | 0.5835 | 0.2204 | 1.0000 | 0.0000 | 0.4912 |
50.1 | 0.5520 | 0.2330 | 0.5446 | 0.2362 | 1.0000 | 0.0000 | 0.5034 |
50.2 | 0.5242 | 0.2454 | 0.5126 | 0.2509 | 1.0000 | 0.0000 | 0.5056 |
50.3 | 0.4957 | 0.2595 | 0.4899 | 0.2626 | 1.0000 | 0.0000 | 0.5030 |
50.4 | 0.4809 | 0.2675 | 0.4789 | 0.2679 | 1.0000 | 0.0000 | 0.5010 |
50.5 | 0.3846 | 0.3344 | 0.3731 | 0.3448 | 1.0000 | 0.0000 | 0.5076 |
50.6 | 0.3717 | 0.3459 | 0.3722 | 0.3456 | 1.0000 | 0.0000 | 0.4997 |
50.7 | 0.3704 | 0.3471 | 0.3714 | 0.3463 | 1.0000 | 0.0000 | 0.4993 |
50.8 | 0.3691 | 0.3483 | 0.3697 | 0.3478 | 1.0000 | 0.0000 | 0.4996 |
50.9 | 0.3676 | 0.3497 | 0.3675 | 0.3498 | 1.0000 | 0.0000 | 0.5001 |
51.0 | 0.3655 | 0.3517 | 0.3651 | 0.3520 | 1.0000 | 0.0000 | 0.5002 |
51.1 | 0.3632 | 0.3538 | 0.3628 | 0.3541 | 1.0000 | 0.0000 | 0.5003 |
51.2 | 0.3610 | 0.3559 | 0.3606 | 0.3562 | 1.0000 | 0.0000 | 0.5002 |
51.3 | 0.3588 | 0.3579 | 0.3585 | 0.3582 | 1.0000 | 0.0000 | 0.5002 |
51.4 | 0.3567 | 0.3599 | 0.3565 | 0.3601 | 1.0000 | 0.0000 | 0.5002 |
51.5 | 0.3547 | 0.3618 | 0.3544 | 0.3621 | 1.0000 | 0.0000 | 0.5002 |
51.6 | 0.3527 | 0.3638 | 0.3523 | 0.3641 | 1.0000 | 0.0000 | 0.5003 |
51.7 | 0.3507 | 0.3657 | 0.3502 | 0.3662 | 1.0000 | 0.0000 | 0.5004 |
51.8 | 0.3485 | 0.3679 | 0.3479 | 0.3685 | 1.0000 | 0.0000 | 0.5004 |
51.9 | 0.3463 | 0.3700 | 0.3456 | 0.3706 | 1.0000 | 0.0000 | 0.5005 |
52.0 | 0.3438 | 0.3725 | 0.3431 | 0.3732 | 1.0000 | 0.0000 | 0.5005 |
52.1 | 0.2350 | 0.4611 | 0.2375 | 0.4549 | 0.0684 | 0.5235 | 0.4345 |
52.2 | 0.1773 | 0.5494 | 0.1806 | 0.5280 | 0.1214 | 0.5781 | 0.3699 |
52.3 | 0.1584 | 0.6095 | 0.1587 | 0.5860 | 0.1413 | 0.5775 | 0.3456 |
52.4 | 0.1368 | 0.6724 | 0.1362 | 0.6461 | 0.1278 | 0.6388 | 0.3413 |
52.5 | 0.1196 | 0.7320 | 0.1204 | 0.7006 | 0.1147 | 0.6768 | 0.3372 |
52.6 | 0.1108 | 0.7769 | 0.1140 | 0.7361 | 0.1077 | 0.7115 | 0.3333 |
Note: ul (l = 1, 2, 3) is normalized by vf2. |
0.070 | 0.924 | 1.010 | 1.120 | 0.024 | 0.025 | 0.025 | 0.003 | 0.002 | 0.002 |
0.150 | 1.233 | 1.305 | 1.372 | 0.044 | 0.045 | 0.044 | 0.022 | 0.030 | 0.022 |
0.190 | 1.399 | 1.482 | 1.550 | 0.060 | 0.060 | 0.060 | 0.034 | 0.044 | 0.037 |
0.200 | 1.439 | 1.524 | 1.593 | 0.061 | 0.061 | 0.061 | 0.038 | 0.047 | 0.041 |
0.210 | 1.484 | 1.574 | 1.641 | 0.062 | 0.062 | 0.062 | 0.041 | 0.050 | 0.045 |
0.368 | 2.344 | 2.496 | 2.595 | 0.063 | 0.062 | 0.062 | 0.030 | 0.036 | 0.034 |
0.370 | 2.354 | 2.507 | 2.607 | 0.063 | 0.062 | 0.062 | 0.031 | 0.038 | 0.035 |
LBV | RF0 | RF1 | |||||||||
0.070 | 5.965 | 6.970 | 7.959 | 5.974 | 6.987 | 7.968 | 5.975 | 6.987 | 7.968 | ||
0.080 | 5.911 | 6.908 | 7.930 | 5.937 | 6.950 | 7.952 | 5.938 | 6.952 | 7.953 | ||
0.090 | 5.848 | 6.829 | 7.897 | 5.895 | 6.880 | 7.937 | 5.895 | 6.881 | 7.938 | ||
0.190 | 5.572 | 6.397 | 7.189 | 5.656 | 6.505 | 7.316 | 5.644 | 6.490 | 7.306 | ||
0.200 | 5.557 | 6.392 | 7.138 | 5.642 | 6.499 | 7.274 | 5.633 | 6.484 | 7.265 | ||
0.210 | 5.567 | 6.401 | 7.106 | 5.638 | 6.488 | 7.236 | 5.627 | 6.480 | 7.230 | ||
0.260 | 5.724 | 6.410 | 7.061 | 5.765 | 6.471 | 7.166 | 5.746 | 6.460 | 7.152 | ||
0.270 | 6.209 | 6.420 | 7.062 | 6.181 | 6.479 | 7.160 | 6.133 | 6.468 | 7.145 | ||
0.280 | 6.264 | 6.439 | 7.070 | 6.214 | 6.491 | 7.163 | 6.168 | 6.479 | 7.148 | ||
0.300 | 6.327 | 6.481 | 7.086 | 6.264 | 6.521 | 7.166 | 6.246 | 6.507 | 7.156 | ||
0.330 | 6.433 | 6.573 | 7.153 | 6.377 | 6.613 | 7.228 | 6.385 | 6.588 | 7.206 | ||
0.360 | 6.560 | 7.067 | 7.260 | 6.500 | 7.043 | 7.326 | 6.490 | 7.001 | 7.280 | ||
0.368 | 6.596 | 7.098 | 7.291 | 6.535 | 7.081 | 7.354 | 6.516 | 7.027 | 7.291 | ||
0.370 | 6.607 | 7.106 | 7.292 | 6.531 | 7.081 | 7.347 | 6.510 | 7.030 | 7.283 |