2.1. Train speeds and blockage ratios in Japanese railways

The train speed $U$ and blockage ratio $R$ ($R=A_{t}/A_{0}$, where $A_{t}$ and $A_{0}$ denote the cross-sectional areas of the train and tunnel, respectively) are key parameters for two trains passing each other. In Japan, the speeds of high-speed (Shinkansen) and conventional trains are less than 320 and 130 km h$^{-1}$, respectively. For typical Shinkansen trains, the blockage ratio is approximately 0.17 (less than 0.2). The blockage ratio of double-decker Shinkansen trains is approximately 0.22; however, their maximum speed is 240 km h$^{-1}$. The blockage ratio of conventional freight trains in a double-track tunnel is less than 0.25, and this ratio is expected to be less than 0.12 for maglev trains.

As the minimum cross-sectional area of the Seikan tunnel is the same as those of other Shinkansen tunnels, the blockage ratios for both Shinkansen and conventional freight trains are less than 0.2.

2.2. Governing equations and exact solutions

Figures 1 and 2 present the schematics of a high-speed train passing a low-speed train in a tunnel. The $x$-axis is set from left to right. The high- and low-speed trains, H and L, run at speeds of $U_{H}$ and $U_{L}$ in the negative and positive $x$-directions, respectively. The noses of two trains meet at $x=0$ and $t=0$, where $t$ denotes time. To simplify the analysis, the blockage ratio and length of both trains are assumed to be equal. Stage A denotes the period after nose–nose passing but before nose–tail passing; stage B indicates the period after nose–tail passing but before tail–tail passing.

Figure 1.

Schematic of two trains passing each other after nose–nose passing but before nose–tail passing.

Figure 2.

Schematic of two trains passing each other after nose–tail passing but before tail–tail passing.

In stage A, two pressure waves, denoted I and II, are considered at the fronts of the two trains. The area between the two waves is divided into six regions: 1, 2, 31, 32, 4 and 5. Region 3, which is occupied by the two running trains, consists of subregions 31 and 32. In region 0, the pressure and density are atmospheric, and the velocity is zero. In stage B, two additional pressure waves, denoted III and IV, are considered in front of each of the two trains, respectively. Regions A1 and A5 represent the states of regions 1 and 5 of stage A, respectively. The area between the two secondary waves is divided into six regions, as in stage A.

As pressure waves I and II are generated around the noses at the moment of nose–nose passing, they propagate through regions 2 and 4, respectively. When they depart from the train tails, the reflected waves return to the noses. In stages A and B, a steady state was assumed after multiple reflections of the waves at the noses and tails for simplicity.

For the following calculations, we assumed a 1-D inviscid compressible flow. In front of and behind each wave, three conservation laws (for continuity, momentum and energy) were adopted. For all other regions, two consecutive regions were connected using the continuity equation, energy equation and isentropic relationship with respect to a coordinate system fixed on each train. For each wave, we have the following relations:

(2.1)\begin{gather} \rho_{f}^{*}(m_{f}-M_{s})=\rho_{b}^{*}\left(m_{b}-M_{s}\right), \end{gather}

(2.2)\begin{gather} X_{f}+\rho_{f}^{*}(m_{f}-M_{s})^{2}=X_{b}+\rho_{b}^{*}\left(m_{b}-M_{s}\right)^{2}, \end{gather}

(2.3)\begin{gather} \frac{\gamma}{\gamma-1}\frac{1/\gamma+X_{f}}{\rho_{f}^{*}}+\frac{1}{2}(m_{f}-M_{s})^{2}=\frac{\gamma}{\gamma-1}\frac{1/\gamma+X_{b}}{\rho_{b}^{*}}+\frac{1}{2}\left(m_{b}-M_{s}\right)^{2}, \end{gather}

where $\rho ^{*}=\rho /\rho _{0}$ is the non-dimensional density, $\rho$ is the density, $\rho _{0}$ is the atmospheric density, $X=p/(\gamma p_{0})$ is the non-dimensional pressure, $p$ is the acoustic pressure, $p_{0}$ is the atmospheric pressure, $\gamma$ is the specific heat ratio, $m=u/c_{0}$ is the non-dimensional velocity, $u$ is the air velocity, $c_{0}$ is the speed of sound, the subscript $f$ indicates the state in front of a wave, the subscript $b$ indicates the state behind a wave, $M_{s}=c_{s}/c_{0}$ and $c_{s}$ is the speed of the acoustic wave (shock wave).

For the two consecutive regions, the continuity, Bernoulli and isentropic equations to be solved are for $i=1,2,32,4$

(2.4)\begin{gather} \rho_{i}^{*}(m_{i}-M_{ref})(1-R_{i})=\rho_{i+1}^{*}(m_{i+1}-M_{ref})(1-R_{i+1}), \end{gather}

(2.5)\begin{gather} \frac{\gamma}{\gamma-1}\frac{1/\gamma+X_{i}}{\rho_{i}^{*}}+\frac{1}{2}(m_{i}-M_{ref})^{2}=\frac{\gamma}{\gamma-1}\frac{1/\gamma+X_{i+1}}{\rho_{i+1}^{*}}+\frac{1}{2}(m_{i+1}-M_{ref})^{2}, \end{gather}

(2.6)\begin{gather} \dfrac{\rho_{i+1}^{*}}{\rho_{i}^{*}}=\left(\frac{1/\gamma+X_{i+1}}{1/\gamma+X_{i}}\right)^{{1}/{\gamma}}, \end{gather}

where $M_{ref}=U_{ref}/c_{0}$ is the Mach number of the reference train, $U_{ref}$ is the reference train speed ($U_{H}$ or $U_{L}$ ), $R_{1}=R_{5}=0$, $R_{2}=R_{4}=R$, $R_{3}=R_{31}=R_{32}=2R$ and $i$ is the index of the region.

The boundary between the subregions 31 and 32 is assumed to be a contact surface. As in $x< u_{3}t$ and $x>u_{3}t$, the airflows are induced by pressure waves I and II, respectively; they can have different density values. Therefore, we have the following continuity conditions for the velocity and pressure, whereas the densities in these two regions are different:

(2.7)\begin{gather} m_{31}=m_{32}=m_{3}, \end{gather}

(2.8)\begin{gather} X_{31}=X_{32}=X_{3}. \end{gather}

The application of (2.1)–(2.8) to regions 1–5 yields 20 variables and 20 equations. This implies that the systems of equations for stages A and B are fully ranked and solvable. It should be noted that flow analysis without pressure waves or a contact surface cannot be solved because the number of equations does not correspond with the number of variables. ‘Python scipy.optimize.minimize’ was used to solve (2.1)–(2.8). Although $R<0.25$ in Japanese double-track tunnels, as outlined in § 2.1, this study considers the range of $0\leq R\leq 0.35$. For this range, the calculation was stable when the initial guesses of $X_{i}$, $m_{i}$, $\rho ^{*}/\rho _{0}-1$ and $M_{s}-1$ were set to zero because flow choking did not occur. Figures 3–5 present the exact solutions for the pressure in each region, calculated using (2.1)–(2.8), where $\alpha =U_{L}/U_{H}$ is the ratio of the speeds of the trains. The difference between $\rho _{31}^{*}$ and $\rho _{32}^{*}$ was negligibly small when $X_{i}\ll 1$.

Figure 3.

Exact solutions of the pressure distribution in each region: $U_{H}=250\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.

Figure 4.

Exact solutions of the pressure distribution in each region: $U_{H}=250\,{\rm km}\,{\rm h}^{-1}$ and $R=0.3$.

Figure 5.

Exact solutions of the pressure distribution in each region: $U_{H}=360\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.

As shown in figures 3–5, the maximum magnitude of pressure is observed in region 3, when $\alpha =1$ in stage A for $R=0.2$ and when $\alpha =0$ in stage B for $R=0.3$. In addition, although the magnitudes of the pressure waves in regions 1 and 5 are almost zero for $U_{H}=250\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$, they have increased (positive and negative) values for larger values of $R$ or $U_{H}$.

2.3. Critical blockage ratio, $R_{c}$

Figures 6 and 7 present the relationship between $p_{3}$, where the magnitude of the pressure reaches its maximum, and $\alpha$. Here, ‘1st order equation’ denotes the approximate solution, explained in § 3.1. For a small value of $R$ (e.g. $R=0.2$), $|p_{3}(\alpha =0)|_{max}<|p_{3}(\alpha =1)|_{max}$, whereas for a large value of $R$ (e.g. $R=0.3$), $|p_{3}(\alpha =0)|_{max}>|p_{3}(\alpha =1)|_{max}$. Here, $|p_{3}|_{max}=\max [|p_{3A}|,|p_{3B}|]$, and $p_{3A}$ and $p_{3B}$ are equal to $p_{3}$ for stages A and B, respectively.

Figure 6.

Relationship between $p_{3}$ and $\alpha$ ($U=250\,{\rm km}\,{\rm h}^{-1}$).

Figure 7.

Relationship between $p_{3}$ and $\alpha$ ($U=360\,{\rm km}\,{\rm h}^{-1}$).

The critical blockage ratio $R_{c}$ is defined as the blockage ratio $R$ that satisfies the following condition:

(2.9)\begin{equation} |p_{3}\left(\alpha=0\right)|_{max}=|p_{3}\left(\alpha=1\right)|_{max}. \end{equation}

Figure 8 presents the exact solution of $R_{c}$, solved numerically. The value of $R_{c}$ decreases with the train speed $U_{H}$. The value of $R_{c}$ is 0.26 when $U_{H}=320\,{\rm km}\,{\rm h}^{-1}$. This indicates that the blockage ratio in conventional high-speed railway tunnels is smaller than $R_{c}$. Consequently, in cases involving mixed traffic with $U_{H}=320\,{\rm km}\,{\rm h}^{-1}$, the conservative worst-case scenario is two trains passing each other at identical speeds, that is, at $\alpha =1$. Maglev trains ($R<0.2$) also satisfy $R< R_{c}$ because $0.18< R_{c}<0.22$ for $500\,{\rm km}\,{\rm h}^{-1}< U_{H}<600\,{\rm km}\,{\rm h}^{-1}$. Most railway tunnels (worldwide) are unlikely to allow the maximum value of $R$ to exceed $R_{c}$; thus, the conservative estimation would be expected to be valid for these tunnels, considering only $\alpha =1$.

Figure 8.

Critical blockage ratios at various speeds of the high-speed train.

3.1. The Hara method

Hara (Hara Reference Hara1961) proposed a theoretical method to evaluate the pressure rise of the compression wave generated by a train entering a tunnel, where the terms of $X^{2}$ are only neglected among the second-order perturbations of the equations representing the compressible airflow, $X^{2}$, $m^{2}$ and $(\rho ^{*}-1)^{2}$. The results of model experiments and field tests show that the formula obtained in this manner effectively predicts the pressure rise. In the method, $m^{2}$ and $(\rho ^{*}-1)^{2}$ are expressed by $X$ and the terms $X^{2}$ are neglected. When we neglect all the second-order perturbations, the system equations become those of linear acoustic theory (Howe Reference Howe1998; Sugimoto & Ogawa Reference Sugimoto and Ogawa1998). In the Hara method, as the terms of the order of $X^{2}$ are neglected, the nonlinearity of pressure is ignored. However, it takes velocity nonlinearities and the velocity–pressure cross-term into account. Indeed, Miyachi (Miyachi Reference Miyachi2019) showed that the Hara method considers the nonlinear source terms.

As $M\sim 0.3$ and $R\sim 0.2$ as described in § 2.1, we consider $M_{H}\ll 1$ and $R\ll 1$. The pressure rise of the compression wave generated by a train entering a tunnel is a weak shock given by $X\sim RM^{2}\sim O[10^{-2}]\ll 1$. Therefore, it is regarded as $O[X_{i}^{2}=R^{2}M^{4}]=O[10^{-4}]$, which can be neglected. Furthermore, because weak shocks satisfy the isentropic relation, $(\rho _{i}/\rho _{0}-1)^{2}=O[X^{2}]$ can also be neglected. While $m\sim O[X=RM^{2}]$ for the air velocity accelerated by a weak shock, $m\sim O[M]$ for the air velocity around the train. Therefore, $m^{2}\sim O[M^{2}]=O[10^{-1}]$ is considered.

The cross-terms between $X$ and higher orders of $M$ complicate the procedure. However, all terms of $X$ are considered, even if the coefficient is a higher order of $M$ (e.g. $M^{2}X$) in the original Hara method. Miyachi et al. (Miyachi et al. Reference Miyachi, Fukuda and Saito2014) applied the Hara method to evaluate the pressure rise of pressure waves generated by a train passing a branch in a tunnel and showed that the results obtained by neglecting the cross-terms for simplicity are in agreement with the experimental results.

Each term of $X_i$ is easily evaluated as $X_{i}\sim RM_{H}^{2}$, assuming the magnitude of the pressure beside a train travelling in a tunnel. This implies that $M_{H}\sim 0.3$, $R\sim 0.2$ and $X_{i}\sim RM_{H}^{2}$ for the train passing problem are the same as the tunnel entering problem. Therefore, the Hara method can be applied to (2.1)–(2.8). For simplicity, the cross-terms $M_{H}X_{i}\sim RM_{H}^{3}\sim O[10^{-3}]$ and $M_{H}^{2}X_{i}\sim RM_{H}^{4}\sim O[10^{-3}]$, respectively, are neglected in §§ 3.3 and 3.4. Therefore, the analysis in this study is based on the $M_{H}$ expansion of $X_{i}$.

3.2. Approximation of governing equations

Here, we introduce an approximation of the governing equations for the approximate equations by neglecting $O[X_{i}^{2}]$. First, we evaluate the equations for each wave. Equations (2.1)–(2.3) yield the following equations:

(3.1)\begin{gather} \dfrac{\gamma(X_{b}-X_{f})}{1+\gamma X_{f}}=\dfrac{2\gamma}{\gamma+1}\Bigg(\left(\dfrac{m_{f}-M_{s}}{c_{f}/c_{0}}\right)^{2}-1\Bigg), \end{gather}

(3.2) \begin{gather} \dfrac{\rho_{b}^{*}}{\rho_{f}^{*}} =\dfrac{m_{f}-M_{s}}{m_{b}-M_{s}}=\dfrac{\left(\gamma-1\right)(1+\gamma X_{f})+(\gamma+1)(1+\gamma X_{b})}{(\gamma+1)(1+\gamma X_{f})+(\gamma-1)(1+\gamma X_{b})}, \end{gather}

where (3.2) is the Rankine–Hugoniot equation.

Equation (3.2) gives

(3.3) \begin{equation} \dfrac{\rho_{b}^{*}}{\rho_{f}^{*}}=\dfrac{m_{f}-M_{s}}{m_{b}-M_{s}}=1+X_{b}-X_{f}+O[X^{2}]=\left(\frac{1/\gamma+X_{b}}{1/\gamma+X_{f}}\right)^{{1}/{\gamma}}. \end{equation}

As this result is well known, the isentropic relationship between the pressure in front of and behind a shock wave is obtained when $O[X_{i}^{2}]$ is neglected. Therefore, we can consider all the flow fields to be isentropic. Thus, we have

(3.4)\begin{equation} \rho_{i}^{*}=\dfrac{\rho_{i}}{\rho_{0}}=\left(\frac{1/\gamma+X_{i}}{1/\gamma}\right)^{{1}/{\gamma}}=1+X_{i},\quad \text{for }i=1,2,31,32,4,5. \end{equation}

Therefore

(3.5) \begin{gather} M_{s}=m_{f}\mp\left(1+\dfrac{\gamma+1}{4}(X_{b}-X_{f})+\dfrac{\gamma-1}{2}X_{f}\right), \end{gather}

(3.6) \begin{gather} m_{b}-m_{f}={\mp}(X_{b}-X_{f}), \end{gather}

where the positive sign denotes the propagation of the pressure wave in the positive $x$-direction.

From (3.4)–(3.6), by neglecting $O[X_{i}^{2}]$, we obtain the following for stage A:

(3.7)\begin{gather} \rho_{1}^{*}=1+X_{1}. \end{gather}

(3.8)\begin{gather} M_{s,{I}}={-}\left(1+\dfrac{\gamma+1}{4}X_{1}\right),\end{gather}

(3.9)\begin{gather} m_{1}={-}X_{1},\end{gather}

and for stage B:

(3.10)\begin{gather} \rho_{1}^{*}=1+X_{1}, \end{gather}

(3.11)\begin{gather} M_{s,{III}}={-}\left(1+\dfrac{\gamma+1}{4}\left(X_{1}+X_{A1}\right)\right), \end{gather}

(3.12)\begin{gather} m_{1}={-}X_{1}, \end{gather}

where $X_{1A}=-m_{1A}$ is used (3.9).

Finally, we can use the same expression $m_{1}=-X_{1}$ for stages A and B, although the values of $X_{1}$ for the two stages are different. Similarly, we also have the following for stages A and B:

(3.13)\begin{equation} m_{5}=X_{5}. \end{equation}

Equations (2.8) and (3.4) yield $\rho _{31}^{*}=\rho _{32}^{*}$ when we neglect $O[X_{i}^{2}]$. Therefore, in this section, we consider the five regions, 1, 2, 3, 4 and 5, shown in figures 1 and 2. The equations for regions 2–4 are also simplified by neglecting $O[X_{i}^{2}]$. Substituting (3.4) into the continuity and Bernoulli equations (2.4) and (2.5) generates the following equations:

(3.14)\begin{gather} (1+X_{i})(m_{i}-M_{ref})(1-R_{i})=(1+X_{i+1})(m_{i+1}-M_{ref})(1-R_{i+1}), \end{gather}

(3.15)\begin{gather} X_{i}+\tfrac{1}{2}(m_{i}-M_{ref})^{2}=X_{i+1}+\tfrac{1}{2}(m_{i+1}-M_{ref})^{2}. \end{gather}

From these results, the following equations are obtained after neglecting $O[X_{i}^{2}]$ for stage A:

Regions 0 and 1:

(3.16)\begin{equation} X_{1}={-}m_{1}. \end{equation}

Regions 1 and 2:

(3.17)\begin{gather} \left(1+X_{1}\right)\left(m_{1}-M_{L}\right)=\left(1+X_{2}\right)\left(m_{2}-M_{L}\right)\left(1-R\right), \end{gather}

(3.18)\begin{gather} X_{1}+\tfrac{1}{2}\left(m_{1}-M_{L}\right)^{2}=X_{2}+\tfrac{1}{2}\left(m_{2}-M_{L}\right)^{2}. \end{gather}

Regions 2 and 3:

(3.19)\begin{gather} \left(1+X_{2}\right)\left(m_{2}+M_{H}\right)\left(1-R\right)=\left(1+X_{3}\right)\left(m_{3}+M_{H}\right)\left(1-2R\right), \end{gather}

(3.20)\begin{gather} X_{2}+\tfrac{1}{2}\left(m_{2}+M_{H}\right)^{2}=X_{3}+\tfrac{1}{2}\left(m_{3}+M_{H}\right)^{2}. \end{gather}

Regions 3 and 4:

(3.21)\begin{gather} \left(1+X_{3}\right)\left(m_{3}-M_{L}\right)\left(1-2R\right)=\left(1+X_{4}\right)\left(m_{4}-M_{L}\right)\left(1-R\right), \end{gather}

(3.22)\begin{gather} X_{3}+\tfrac{1}{2}\left(m_{3}-M_{L}\right)^{2}=X_{4}+\tfrac{1}{2}\left(m_{4}-M_{L}\right)^{2}. \end{gather}

Regions 4 and 5:

(3.23)\begin{gather} \left(1+X_{4}\right)\left(m_{4}+M_{H}\right)\left(1-R\right)=\left(1+X_{5}\right)\left(m_{5}+M_{H}\right), \end{gather}

(3.24)\begin{gather} X_{4}+\tfrac{1}{2}\left(m_{4}+M_{H}\right)^{2}=X_{5}+\tfrac{1}{2}\left(m_{5}+M_{H}\right)^{2}. \end{gather}

Regions 5 and 0:

(3.25)\begin{equation} X_{5}=m_{5}. \end{equation}

We obtain the equations for stage B by replacing $M_{L}$ with $-M_{H}$ and $M_{H}$ with $-M_{L}$ in (3.16)–(3.25). Although the speeds of the pressure waves differ for stages A and B ($c_{s,{I}}\neq c_{s,{III}}$), they do not explicitly appear in (3.16)–(3.25).

3.3. First-order equations (equations of $O[M_{H}^{2}]$)

The approximations of (3.16)–(3.25) when $O[X_{i}^{2}]$ and $O[M_{H}^{3}]$ are neglected for stage A are as follows (see Appendix B):

(3.26)\begin{gather} c_{p1}=\frac{R^{2}}{\left(1-R\right)\left(1-2R\right)}(1-\alpha^{2})\geq0, \end{gather}

(3.27)\begin{gather} c_{p5}={-}c_{p1}\leq0, \end{gather}

(3.28)\begin{gather} c_{p2}=\frac{R^{2}}{\left(1-R\right)\left(1-2R\right)}-\frac{R\left(2-\left(4-R\right)R\right)}{\left(1-R\right)^{2}\left(1-2R\right)}\alpha^{2}, \end{gather}

(3.29)\begin{gather} c_{p4}=\frac{R^{2}}{\left(1-R\right)\left(1-2R\right)}\alpha^{2}-\frac{R\left(2-\left(4-R\right)R\right)}{\left(1-R\right)^{2}\left(1-2R\right)}, \end{gather}

(3.30)\begin{gather} c_{p3}={-}\left[\frac{R}{1-R}\left(1+\alpha\right)^{2}+\frac{R\left(1-R\right)}{\left(1-2R\right)^{2}}\left(1-\alpha\right)^{2}\right]. \end{gather}

As the relative errors of (3.26)–(3.30) denote the order of $M_{H}$ (see Appendix B), these equations are first-order equations. Owing to the symmetry of the pressure fields between stages A and B, as shown in figures 1 and 2, replacing $M_{L}$ with $-M_{H}$ and $M_{H}$ with $-M_{L}$ in (3.26)–(3.30) yields the equations for stage B. Thus, we have shown that $c_{p3}$ for stage B is the same as that for stage A. Similarly, $c_{p1}$ for stage B is the same as $c_{p5}$ for stage A.

From (3.26)–(3.30), we obtain the following pressure differences between two consecutive regions:

(3.31)\begin{gather} \Delta c_{p1}=\frac{X_{1}-0}{\tfrac{1}{2}M_{H}^{2}}=\frac{R^{2}}{\left(1-R\right)\left(1-2R\right)}(1-\alpha^{2}), \end{gather}

(3.32)\begin{gather} \Delta c_{p21}=\frac{\Delta X_{21}}{\tfrac{1}{2}M_{H}^{2}}=\frac{X_{2}-X_{1}}{\tfrac{1}{2}M_{H}^{2}}={-}\frac{R\left(2-R\right)}{\left(1-R\right)^{2}}\alpha^{2}, \end{gather}

(3.33)\begin{gather} \Delta c_{p32}=\frac{\Delta X_{32}}{\tfrac{1}{2}M_{H}^{2}}=\frac{X_{3}-X_{2}}{\tfrac{1}{2}M_{H}^{2}}={-}\frac{R\left(2-3R\right)}{\left(1-2R\right)^{2}}\left(1-\frac{R}{1-R}\alpha\right)^{2}, \end{gather}

(3.34)\begin{gather} c_{p3}=\frac{X_{3}}{\tfrac{1}{2}M_{H}^{2}}=\Delta c_{p1}+\Delta c_{p21}+\Delta c_{p32}. \end{gather}

Figures 9–11 present the relationships between $\alpha$ and $\Delta c_{p1}$, $\Delta c_{p21}$, $\Delta c_{p32}$ and $c_{p3}$ for various values of $R$ for stage A.

Figure 9.

Relationships between pressure coefficients and $\alpha$ for $R=0.2$ (first-order equations).

Figure 10.

Relationships between pressure coefficients and $\alpha$ for $R=0.3$ (first-order equations).

Figure 11.

Relationships between pressure coefficients and $\alpha$ for $R=0.35$ (first-order equations).

In (3.31), $\Delta c_{p1}$ denotes the pressure rise of the pressure wave in front of train H. In (3.32), $\Delta c_{p21}$ denotes the pressure drop beside train L (region 2) when it runs alone through incompressible air before passing train H (see Appendix C). In (3.33), $\Delta c_{p32}$ denotes the pressure difference between regions 2 and 3 in an incompressible flow. Therefore, in the first-order equation, the pressure field around the two trains is the linear superposition of that of the incompressible flow and the pressure variation generated by the pressure waves. As $X_{1}+X_{5}=0$, the impacts of the two pressure waves cancel out in region 3, and the two waves do not have any impact on this region. Moreover, in the first order of $R$, we have $X_{3}\sim X_{2}+X_{4}\sim -RM_{H}^{2}(1+\alpha ^{2})$. This means that the pressure between two passing trains is the approximate superposition of the pressure surrounding each train running alone.

The term $\Delta X_{21}$ is proportional to $\alpha ^{2}M_{H}^{2}$, which is the dynamic pressure of the low-speed train L. For $0\leq \alpha \leq 1$ and $0\leq R\leq 1/2$, (3.32) indicates that $\Delta c_{p21}$ is a decreasing function of $\alpha$ and $\Delta c_{p21}\leq 0$, implying that $|\Delta c_{p21}|$ is an increasing function of $\alpha$ (as shown in figures 9–11).

The term $\Delta X_{32}$ is proportional to $m_{2H}^{2}=(1-({R}/({1-R}))\alpha )^{2}M_{H}^{2}$, where $m_{2H}=(1-({R}/({1-R}))\alpha )M_{H}$ is the relative air velocity of region 2 in the coordinate system fixed on train H. Here, $m_{2}=-({R}/({1-R}))\alpha M_{H}$ is the speed of the airflow in region 2 induced by train L for incompressible flow. This implies that train L induces the airflow that follows train H with the speed $({R}/({1-R}))\alpha M_{H}$; this airflow that follows train H reduces the relative speed of train H and the magnitude of the pressure variation for the two trains passing each other. The effect of this reduction is more apparent as $\alpha$ increases; no following wind is observed for $\alpha =0$. Equation (3.33) indicates that the vertex of $\Delta c_{p32}$ is located at $\alpha ={1}/{R}-1\geq 1$ for $0\leq R\leq 1/2$. Therefore, for $0\leq \alpha \leq 1$ and $0\leq R\leq 1/2$, $\Delta c_{p32}$ is an increasing function of $\alpha$, and $\Delta c_{p32}<0$. Thus, $|\Delta c_{p32}|$ is a decreasing function of $\alpha$ (as shown in figures 9–11).

The term $X_{1}$ is proportional to $(1-\alpha ^{2})M_{H}^{2}$, which is the difference in dynamic pressure between the two trains. Therefore, when the two trains pass each other at the same speed, $X_{1}=0$, that is, the compression wave generated by train H cancels out the expansion wave generated by train L in front of train H. For $0\leq \alpha \leq 1$ and $0\leq R\leq 1/2$, (3.31) indicates that $\Delta c_{p1}$ is a decreasing function of $\alpha$ and $\Delta c_{p1}\geq 0$. Figures 9–11 indicate that this term is relatively small compared with $\Delta c_{p21}$ and $\Delta c_{p32}$. For incompressible flow, $c_{p3}=\Delta c_{p21}+\Delta c_{p32}<0$. However, the compressibility contributes to $|c_{p3}|$ through $\Delta c_{p1}$.

As $\Delta c_{p21}$ is a decreasing function of $\alpha$, $\Delta c_{p21}(\alpha =0)>\Delta c_{p21}(\alpha =1)$, whereas $\Delta c_{p32}$ is an increasing function of $\alpha$, and $\Delta c_{p32}(\alpha =0)<\Delta c_{p32}(\alpha =1)$. The magnitude of $\Delta c_{p1}$ is relatively small compared with those of $\Delta c_{p21}$ and $\Delta c_{p32}$. For a small value of $R$, as ${\rm d}\Delta c_{p21}/{\rm d}\alpha \sim -2R\alpha ^{2}$ is the dominant term, $c_{p3}$ is a decreasing function, whereas for a large value of $R$, as ${\rm d}\Delta c_{p32}/{\rm d}\alpha \sim 4R^{2}$ is also considerable, $c_{p3}$ is an increasing function around $\alpha \sim 0$. The critical blockage ratio $R_{c}$, where the dominant term changes, satisfies the following equation:

(3.35)\begin{equation} c_{p3}\left(\alpha=0\right)=c_{p3}\left(\alpha=1\right). \end{equation}

By solving the first-order equations (3.31)–(3.35), we have $R_{c}=(\sqrt {3}-1)/(2\sqrt {3}-1)=0.297\sim 0.3$. Figure 8 presents a comparison between the first-order and exact solutions of $R_{c}$. The first-order solution is constant as $U_{H}$ changes, whereas the exact solution decreases with $U_{H}$. A coarse estimation of the maximum magnitude of the pressure induced by two trains passing through a tunnel is as follows:

(3.36)\begin{gather} \left|p_{3}\left(\alpha=1\right)\right|=\left|-\frac{1}{2}\rho_{0}U_{H}^{2}\frac{4R}{1-R}\right|,\quad \text{for }R\leq0.3, \end{gather}

(3.37) \begin{gather} \left|p_{3}\left(\alpha=0\right)\right|=\left|-\frac{1}{2}\rho_{0}U_{H}^{2}\frac{R}{1-R}\Bigg(1+\left(\frac{1-R}{1-2R}\right)^{2}\Bigg)\right|,\quad \text{for }R\geq0.3. \end{gather}

3.4. Second-order equations (equations of $O[M_{H}^{3}]$)

The approximate equations of (3.16)–(3.25) when $O[X_{i}^{2}]$ and $O[M_{H}^{4}]$ are neglected for stage A are as follows (see Appendix B):

(3.38) \begin{align} c_{p1A}^{\left(2\right)}&= \frac{R^{2}}{\left(1-2R\right)\left(1-R\right)}(1-\alpha^{2}) \nonumber\\ &\quad -\frac{R^{2}}{1-R}\frac{3}{4}M_{H}\left(\left(1+\alpha\right)^{3}+\frac{\frac{4}{3}R^{2}}{\left(1-R\right)\left(1-2R\right)^{2}}\left(1+\alpha\right)^{2}\left(1-\alpha\right)\right.\nonumber\\ &\quad -\left.\frac{1}{1-2R}\left(1+\alpha\right)\left(1-\alpha\right)^{2}\vphantom{\frac{\dfrac{4}{3}R^{2}}{\left(1-R\right)\left(1-2R\right)^{2}}}\right),\end{align}

(3.39)\begin{align}c_{p5A}^{\left(2\right)}&={-}\frac{R^{2}}{\left(1-2R\right)\left(1-R\right)}(1-\alpha^{2})\nonumber\\ &\quad -\frac{R^{2}}{1-R}\frac{3}{4}M_{H}\left(\left(1+\alpha\right)^{3}-\frac{\frac{4}{3}R^{2}}{\left(1-R\right)\left(1-2R\right)^{2}}\left(1+\alpha\right)^{2}\left(1-\alpha\right)\right.\nonumber\\ &\quad -\left.\frac{1}{1-2R}\left(1+\alpha\right)\left(1-\alpha\right)^{2}\vphantom{\frac{\dfrac{4}{3}R^{2}}{\left(1-R\right)\left(1-2R\right)^{2}}}\right), \end{align}

(3.40)\begin{align} c_{p3A}^{\left(2\right)}&={-}\left[\frac{R}{1-R}\left(1+\alpha\right)^{2}+\frac{\left(1-R\right)R}{\left(1-2R\right)^{2}}\left(1-\alpha\right)^{2}\right]\nonumber\\ &\quad -\frac{R^{2}}{1-R}\frac{3}{4}M_{H}\left(1+\alpha\right)^{3}+\frac{R^{2}}{\left(1-2R\right)^{3}}\left(\frac{1}{1-R}-\frac{4}{3}R\right)\nonumber\\ &\quad \times \frac{3}{4}M_{H}\left(1+\alpha\right)\left(1-\alpha\right)^{2}, \end{align}

(3.41)\begin{align} c_{p2A}^{\left(2\right)}-c_{p1A}^{\left(2\right)}&={-}\frac{R\left(2-R\right)}{\left(1-R\right)^{2}}\alpha^{2}-\frac{R^{3}\left(2-R\right)}{\left(1-2R\right)\left(1-R\right)^{3}}M_{H}\alpha(1-\alpha^{2}), \end{align}

(3.42)\begin{align} c_{p4A}^{\left(2\right)}-c_{p5A}^{\left(2\right)}&={-}\frac{R\left(2-R\right)}{\left(1-R\right)^{2}}+\frac{R^{3}\left(2-R\right)}{\left(1-2R\right)\left(1-R\right)^{3}}M_{H}(1-\alpha^{2}), \end{align}

where $c_{piA}^{(2)}$ is the pressure coefficient in region i when considering $O[M_{H}^{3}]$ for stage A. As the relative errors of (3.38)–(3.42) denote the order of $O[M_{H}^{2}]$, these equations are second-order equations. Replacing $M_{L}$ with $-M_{H}$ and $M_{H}$ with $-M_{L}$ in these equations, we obtain the results for stage B, e.g.

(3.43)\begin{align} c_{p3B}^{\left(2\right)} & ={-}\left[\frac{R}{1-R}\left(1+\alpha\right)^{2}+\frac{\left(1-R\right)R}{\left(1-2R\right)^{2}}\left(1-\alpha\right)^{2}\right]\nonumber\\ &\quad +\frac{R^{2}}{1-R}\frac{3}{4}M_{H}\left(1+\alpha\right)^{3}-\frac{R^{2}}{\left(1-2R\right)^{3}}\left(\frac{1}{1-R}-\frac{4}{3}R\right)\frac{3}{4}M_{H}\left(1+\alpha\right)\left(1-\alpha\right)^{2}. \end{align}

The difference between the pressure in region 3 at stages A and B is as follows:

(3.44)\begin{align} \Delta c_{p3BA}^{\left(2\right)}& \equiv\Delta c_{p3B}^{\left(2\right)}-\Delta c_{p3A}^{\left(2\right)}\nonumber\\ & =\left[\frac{1}{1-R}\left(1+\alpha\right)^{3}-\frac{1}{\left(1-2R\right)^{3}}\left(\frac{1}{1-R}-\frac{4}{3}R\right)\left(1+\alpha\right)\left(1-\alpha\right)^{2}\right]\frac{3R^{2}M_{H}}{2}, \end{align}

(3.45)\begin{gather} \Delta c_{p3BA}^{\left(2\right)}\left(\alpha=0\right)={-}\frac{R^{2}M_{H}}{2}\dfrac{2R\Big(12\left(R-\frac{2}{3}\right)^{2}+\frac{5}{3}\Big)}{\left(1-R\right)\left(1-2R\right)^{3}}, \end{gather}

(3.46)\begin{gather} \Delta c_{p3BA}^{\left(2\right)}\left(\alpha=1\right)=\frac{R^{2}M_{H}}{2}\frac{24}{1-R}. \end{gather}

Figure 12 presents the values of $\Delta c_{p3BA}^{(2)}/M_{H}$ for $R=0.2$ and $0.3$. For $0\leq R\leq 1/2$, $\Delta c_{p3BA}^{(2)}$ is an increasing function of $\alpha$; and here, $\Delta c_{p3BA}^{(2)}(\alpha =0)<0$ and $\Delta c_{p3BA}^{(2)}(\alpha =1)>0$. Therefore, $c_{p3B}^{(2)}(\alpha =0)< c_{p3A}^{(2)}(\alpha =0)<0$ and $c_{p3A}^{(2)}(\alpha =1)< c_{p3B}^{(2)}(\alpha =1)<0$. The maximum magnitude of the pressure as the two trains pass each other is observed in stages A and B for $\alpha =1$ and $0$, respectively, regardless of the value of $R$, as shown in figures 3–5.

Figure 12.

Pressure difference between stages A and B in region 3 (second-order equations).

Similarly, the pressure variations induced by pressure waves I and III, $\Delta c_{p,{I}}$ and $\Delta c_{p,{III}}$, respectively, are expressed as follows:

(3.47)\begin{align} \Delta c_{p,\textrm{I}}^{\left(2\right)}&=c_{p1A}^{\left(2\right)}-0 \nonumber\\ & =\frac{R^{2}}{\left(1-2R\right)\left(1-R\right)}(1-\alpha^{2}) \nonumber\\ &\quad -\frac{R^{2}}{1-R}\frac{3}{4}M_{H}\left(\left(1+\alpha\right)^{3}+\frac{\tfrac{4}{3}R^{2}}{\left(1-R\right)\left(1-2R\right)^{2}}\left(1+\alpha\right)^{2}\left(1-\alpha\right)\right.\nonumber\\ &\quad-\left.\frac{1}{1-2R}\left(1+\alpha\right)\left(1-\alpha\right)^{2}\vphantom{\frac{\tfrac{4}{3}R^{2}}{\left(1-R\right)\left(1-2R\right)^{2}}}\right), \end{align}

(3.48)$$\begin{align} \Delta c_{p,{III}}^{\left(2\right)}&=\Delta c_{p1B}^{\left(2\right)}-c_{p1A}^{\left(2\right)}={-}2\frac{R^{2}}{\left(1-2R\right)\left(1-R\right)}(1-\alpha^{2})\nonumber\\ &\quad +\frac{R^{2}}{1-R}\frac{3}{2}M_{H}\left(\left(1+\alpha\right)^{3}-\frac{1}{1-2R}\left(1+\alpha\right)\left(1-\alpha\right)^{2}\right). \end{align}$$

Figures 13–16 present the pressure variations induced by pressure waves I and III. In the first-order equations, pressure waves I and III are always compressive and expansive, respectively. In particular, for $\alpha =1$, the pressure variations of these waves are zero. However, in the exact solutions or in the second-order equations, the variations are zero at $\alpha =\alpha _{cri}<1$, where $\alpha _{cri}$ is the value of the zero pressure variation. As shown in the exact solutions in figures 13–16, $\alpha _{cri}$ is close to one. Solving each of $\Delta c_{p,{I}}^{(2)}=0$ and $\Delta c_{p,{III}}^{(2)}=0$ in the expanded form with $1-\alpha$ and $R$ yields the same solution as follows:

(3.49)\begin{equation} \alpha_{cri}=\dfrac{2+6R+3\left(1+R\right)M}{2+6R+9\left(1+R\right)M}. \end{equation}

For $\alpha >\alpha _{cri}$, pressure waves I and III are expansive and compressive, respectively, and vice versa. Figures 13–16 also show that the values of $\alpha _{cri}$ and the exact solutions are in good agreement.

Figure 13.

Pressure variations induced by pressure wave I: $U_{H}=250\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.

Figure 14.

Pressure variations induced by pressure wave III: $U_{H}=250\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.

Figure 15.

Pressure variations induced by pressure wave I: $U_{H}=360\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.

Figure 16.

Pressure variations induced by pressure wave III: $U_{H}=360\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.

Replacing $M_{H}$ and $M_{L}$ with $M_{L}$ and $M_{H}$, respectively, in (3.47) and (3.48) yields $\Delta c_{p,{II}}$ and $\Delta c_{p,{IV}}$. Figures 17 and 18 show the pressure variation induced by pressure waves II and IV. Figures 17 and 18 indicate that pressure waves II and IV are always expansive and compressive, respectively. It is easily shown that $\Delta c_{p,{I}}(\alpha =1)=\Delta c_{p,{II}}(\alpha =1)$ and $\Delta c_{p,{III}}(\alpha =1)=\Delta c_{p,{IV}}(\alpha =1)$.

Figure 17.

Pressure variations induced by pressure waves II and IV: $U_{H}=250\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.

Figure 18.

Pressure variations induced by pressure waves II and IV: $U_{H}=360\,{\rm km}\,{\rm h}^{-1}$ and $R=0.2$.