4.1. Pressure and density relationships in the compressed gas

To better represent the quantities in different zones, the gas density and velocity evolutions in figures 3(a) and 3(b) are sketched in figure 4. The evolutions of the particle volume fraction and the particle mean density are shown in figures 3(c) and 3(d). As illustrated in figure 1(b) and (c), after the interaction of the shock with the gas–particle contact surface, a compressed gas zone (2) is generated. From a modelling point of view, it is important to derive relationships between the pure gas zone (1) and the compressed gas zone (2), as follows:

(4.1a–c)\begin{equation} p_{g,1} < p_{g,2}, \quad \rho_{g,1} < \rho_{g,2}, \quad u_{g,1} > u_{g,2}. \end{equation}

Similarly, between the compressed gas zone (2) and the post-shock gas–particle mixture (3), one has the following:

(4.2a–c)\begin{equation} p_{g,2} = p_{g,3}, \quad \rho_{g,2} > \rho_{g,3} \quad u_{g,2} = u_{g,3}. \end{equation}

Figure 4.

Sketch of gas properties after the shock–particle cloud interaction: (a) density and (b) gas velocity evolutions.

With regard to the gas density distribution, and as depicted in figure 4(a), the gas density inside the post-shock gas–particle mixture $\rho _{g,3}$ might be higher than, lower than or equal to the density of the pure gas zone $\rho _{g,1}$. One can use $\rho _{g,3l}$, $\rho _{g,3c}$ and $\rho _{g,3r}$ to denote the three different cases. The numerical results show that, when one has $\rho _{g,1}<\rho _{g,3l}$ (for weak $M_s$), the reflected pressure expansion tends to propagate towards the $x^-$ direction, while if $\rho _{g,1}>\rho _{g,3r}$ (for strong $M_s$), the pressure expansion propagates inversely for most numerical simulations.

The gas velocity evolution, described in figure 4(b), indicates that, during the interaction, one always has $u_{g,1}>u_{g,2}$, $u_{g,2} = u_{g,3}$ and $u_{g,3}>u_{g,4}=0$. It is interesting to note that, for a wide range of particle volume fractions ($\tau _{v,0}\approx {O}(10^{-5}\text {--}10^{-3})$), the velocity ratio in the pure and in the compressed gas regions is almost constant over a wide range of Mach numbers ($1<M_s<4$) (see figure 5). We assume therefore that

(4.3)\begin{equation} \frac{u_{g,2}}{u_{g,1}}\approx f(\tau_{v,0}), \end{equation}

where $\tau _{v,0}$ is the initial particle volume fraction, which is considered to be the main factor influencing the variation of $u_{g,2}/u_{g,1}$. Equations (4.1a–c) to (4.3) are considered as the fundamental hypothesis of the reduced-order modelling.

Figure 5.

Velocity ratio $u_{g,2}/u_{g,1}$ over a wide range of incident shock Mach numbers, for $D = 1\ \mathrm {\mu } \textrm {m}$ and: $\tau _{v,0} = 5.2\times 10^{-5}$ (green), $\tau _{v,0} = 5.2\times 10^{-4}$ (blue) and $\tau _{v,0} = 5.2\times 10^{-3}$ (blue-green).

5.1. Velocity of the compressed gas

It is noted from various numerical simulations that, for weak initial Mach numbers, the reflected pressure expansion has the velocity of the sound speed in zone (1). Here we consider the particular case where $\rho _{g,1}=\rho _{g,3c}$, as illustrated in figure 4(a). In this case, the interface between zone (1) and zone (2) remains stationary in the laboratory frame, which can be characterized by $u_{g,1} = c_1= \sqrt {{\gamma p_1}/{\rho _{g,1}}}$, with $c_1$ being the sound speed in zone (1). Using the gas properties across the shock wave (White Reference White2011), one can deduce that

(5.5)\begin{equation} c_0\frac{2}{\gamma+1} \frac{M_s^2 - 1}{M_s} = \sqrt{\frac{\gamma p_1}{\rho_{g,1}}}, \end{equation}

which can be simplified to

(5.6)\begin{equation} (M_s^2-1)^2 = \left(\frac{\gamma + 1}{2}\right)^2 \mathcal{F}_1(M_s,\gamma)\, \mathcal{F}_2(M_s,\gamma). \end{equation}

The Mach number that satisfies (5.6) is known as the critical Mach number, and is $M_{s,cr} = 2.0$ for a monatomic ideal gas ($\gamma = 7/5$). The results of figure 3(a) show that $\rho _{g,3}=\rho _{g,1}$ and $u_{g,3}=u_{g,2}$ when $M_s=M_{s,cr}$. The conservation of kinetic energy before and after the passage of the shock gives

(5.7)\begin{equation} \rho_{g,1}u_{g,1}^2 = \tau_{v,0}\rho_p{u}_{g,3}^2 + (1-\tau_{v,0})\rho_{g,3}{u}_{g,3}^2, \end{equation}

where ${u}_{g,3}$ is the velocity of the shocked gas in the gas–particle mixture and $\rho _p$ is the mass density of the particles. By combining (4.3) and (5.7), the estimate of $u_{g,2}$ can be obtained for a given $\tau _{v,0}$ as

(5.8)\begin{equation} \frac{{u}_{g,2}}{u_{g,1}} \approx \left(\frac{{u}_{g,2}}{u_{g,1}}\right)_{cr} = \sqrt{\frac{1}{1-\tau_{v,0}+\tau_{v,0}\dfrac{\rho_p}{\rho_{g,1}}}}. \end{equation}

5.2. Density of the compressed gas

To estimate the density of the compressed gas $\rho _{g,2}$, one can assume that the pressure wave reflection process obeys an isentropic expansion. The isentropic hypothesis is discussed in appendix A. The conservation of momentum leads to

(5.9)\begin{equation} \rho_{g,2}u_{g,2}^2 + p_1\left(\frac{\rho_{g,2}}{\rho_{g,1}} \right)^\gamma = \rho_{g,1}u_{g,1}^2 + p_1. \end{equation}

For a given estimated $u_{g,2}$, the solution of (5.9) can provide an estimate of $\rho _{g,2}$. A Newton–Raphson method is used to solve (5.9).

The estimate of $\rho _{g,2}$ can also contribute to the evaluation of the intensity of the pressure expansion. Moreover, knowing the two properties of the flow, $u_{g,2}$ and $\rho _{g,2}$, one can easily predict the propagation direction of the pressure expansion after the interaction of the shock with the spray by using the criterion given by (5.1).

5.3. Spray dispersion

Equation (5.7) can also be applied to estimate the particle dispersion in the post-shock gas–particle mixture (3). Figure 6 shows the configuration in the proximity of the transmitted shock inside the gas–particle zone in the shock-attached frame, where the properties across the transmitted shock wave are depicted in figure 6. One knows that

(5.10a–c)\begin{equation} {p}_3={p}_2, \quad {u}_{g,3}= {u}_{g,2}, \quad u_{g,4} = 0, \end{equation}

where ${u}_{g,3}$ and ${u}_{g,2}$ are estimated quantities. Thus, the conservation of mass across the shock wave gives

(5.11)\begin{equation} {\rho}_{g,3} (V_s- {u}_{g,2}) = {\rho}_{g,4}V_s. \end{equation}

Figure 6.

Sketch of the shock front transmitted into the gas–particle zone in the shock-attached frame. Here CG = compressed gas, P = particles, and G = unshocked gas.

Taking into account the initial volume fraction $\tau _{v,0}$ of particles, the conservation of momentum gives

(5.12)\begin{align} p_2 + {\rho}_{g,3}(1-\tau_v)(V_s- {u}_{g,2})^2 + \rho_p\tau_v(V_s- {u}_{g,2})^2 = p_4 + {\rho}_{g,4}(1-\tau_{v,0})V_s^2 +\rho_p\tau_{v,0}V_s^2, \end{align}

where $\tau _v$ is the volume fraction of the particles in the post-shock region (3). Before the shock passage, the initial cloud length is $V_s t$ and becomes $(V_s - {u}_{g,2} ) t$ afterwards. Therefore, the post-shock volume fraction of the particles ${\tau _v}$ can be linked to the pre-shock volume fraction $\tau _{v,0}$ through

(5.13)\begin{equation} {\tau_v} = \tau_{v,0} \,\frac{V_s}{V_s - {u}_{g,2}} {.} \end{equation}

By combining (5.11)–(5.13), one can deduce an analytical expression for the velocity of the transmitted shock wave $V_s$:

(5.14)\begin{equation} V_s = \frac{p_2 - p_4}{(\rho_{g,4}+\rho_p\tau_{v,0}) {u}_{g,2}}. \end{equation}

Knowing $V_s$, the volume fraction of the particles in region (3) can be calculated by (5.13).

5.4. Assessment of the reduced-order model

From the above discussion, the modelling is achieved for $u_{g,2}$, $\rho _{g,2}$, $p_2$ and $\tau _v$, for a given initial volume fraction $\tau _{v,0}$ and incident Mach number $M_s$. The estimates of $\rho _{g,2}$ and $\tau _v$ rely especially on the accuracy of $u_{g,2}$. However, the post-shock maximal particle volume fraction has similar values for particles of different diameters. The maximal volume fractions of small particles can provide guideline values for the larger ones. The assessment of the proposed model, especially for the estimation of $u_{g,2}$, $\rho _{g,2}$ and $\tau _v$, is achieved through a direct comparison with the results from the NS solver.

Taking an example of the initial particle volume fraction $\tau _{v,0} = 5.2\times 10^{-4}$, in critical conditions, in which $\rho _{g,3c} = \rho _{g,1}$, the velocity ratio calculated by the NS code is $u_{g,2}/u_{g,1}=0.9$, and (5.7) gives a ratio of $0.93$. If the gas velocity $u_{g,2}$ in the compressed zone is correctly estimated, the model evaluations for $\rho _{g,2}$ and $p_2$ are in good agreement with the numerical simulations, as shown in figure 7(a–d). For $\tau _{v,0} = 5.2\times 10^{-4}$ and $M_s = 2.0$, the density estimated by (5.9) is $\rho _{g,2} = 3.4\ \textrm {kg}\,\textrm {m}^{-3}$, and the value calculated by the NS code is $\rho _{g,2} = 3.44\ \textrm {kg}\,\textrm {m}^{-3}$.

Figure 7.

Evolution of different flow properties as a function of $M_s$: (a) gas velocity, (b) gas density, (c) gas pressure and (d) particle volume fraction. Simulation using NS code: $\tau _{v,0} =5.2\times 10^{-3}$ (yellow dot-dashed); $\tau _{v,0} =5.2\times 10^{-4}$ (red dashed); and $\tau _{v,0} =5.2\times 10^{-5}$ (blue-green dotted). Current model: $\tau _{v,0} =5.2\times 10^{-3}$ (inverted triangles); $\tau _{v,0} =5.2\times 10^{-4}$ (circles); and $\tau _{v,0} =5.2\times 10^{-5}$ (triangles).

For the ratio of the particle volume fraction $\tau _v/\tau _{v,0}$, using (5.13), one can have higher relative differences compared to the numerical simulation, since the estimation of this ratio is based on the modelling of both $u_{g,2}$ and $\rho _{g,2}$. In the case where these two parameters have a relative difference of $5\,\%$, one may have a relative difference of $28\,\%$ for $\tau _v/\tau _{v,0}$ as a result of the difference accumulation, as shown in figure 7(d).

The analytical model is assessed in this study by the NS solver for the interaction between supersonic flows of Mach number $M_s=1.1\text {--}4$ and a particle cloud of volume fraction $\tau _{v,0}=10^{-5}\text {--}10^{-3}$ and of particle diameters $D=1\text {--}10\ \mathrm {\mu } \textrm {m}$. For shock waves of Mach number higher than $M_s=5$, the heat transfer induced by the hypersonic shock should be considered. The present model can be applied to solid particles or liquid droplets having relatively high surface tension values. For higher volume fraction $\tau _{v,0} > 10^{-3}$ of particle clouds, the interactions among the particles become important, such as particle collision and coalescence of droplets, as noted by Elghobashi (Reference Elghobashi1994). For water droplets of diameter larger than $D > 10\ \mathrm {\mu } \textrm {m}$, the breakup of the droplets in high-velocity gas flow becomes important. Particles of diameter ${O}(10\ \mathrm {\mu } \textrm {m})$ can be regarded as stable fragments of larger droplets according to Pilch & Erdman (Reference Pilch and Erdman1987).

5.5. Influence of the particle response time $\tau _p$

The particle response time scale $\tau _p$ is defined to describe the response ability of the particles to the carrier flow movement, which can have a simple expression:

(5.15a,b)\begin{equation} \tau_p = \frac{\rho_pD^2}{18\mu_g}\frac{1}{\varPhi(Re_p)},\quad \varPhi(Re_p) = 1.0+0.15Re_p^{0.687}. \end{equation}

Here $\rho _p$ is the mass density of the particles, $D$ is the diameter of the particles, $\mu _g$ is the dynamic viscosity of air, and $Re_p$ is the particle Reynolds number.

Let us consider a shock wave of $M_s=1.1$ interacting with a cloud of particles. The influence of the particle response time on resulting flow evolution is shown in figure 8 at $t=600\ \mathrm {\mu } \textrm {s}$ after the start of the interaction.

Figure 8.

Evolution with distance of normalized (a) particle volume fraction, (b) gas pressure, (c) gas velocity and (d) particle velocity, at $t = 600\ \mathrm {\mu } \textrm {s}$, $M_s = 1.1$, $\tau _{v,0}=5.2\times 10^{-4}$, $p_0 = 1.013\ \textrm {bar}$ and $u_{g,0}=55.19\ \textrm {m}\,\textrm {s}^{-1}$, for different particle diameters. Curves: $D=1\ \mathrm {\mu } \textrm {m}$, $\tau _p=2.3\ \mathrm {\mu } \textrm {s}$ (dark blue); $D=2\ \mathrm {\mu } \textrm {m}$, $\tau _p=7.7\ \mathrm {\mu } \textrm {s}$ (blue-green); $D=4\ \mathrm {\mu } \textrm {m}$, $\tau _p=25\ \mathrm {\mu } \textrm {s}$ (blue); $D=6\ \mathrm {\mu } \textrm {m}$, $\tau _p=49\ \mathrm {\mu } \textrm {s}$ (green); $D=8\ \mathrm {\mu } \textrm {m}$, $\tau _p=78\ \mathrm {\mu } \textrm {s}$ (orange); $D=10\ \mathrm {\mu } \textrm {m}$, $\tau _p=0.1\ \textrm {ms}$ (red); and original gas–particle contact surface (black dashed).

Figure 8(a) shows the evolution of the particle volume fractions for different particle diameters varying from $1\ \mathrm {\mu } \textrm {m}$ to $10\ \mathrm {\mu } \textrm {m}$, with particle response time varying from $2.3\times 10^{-6}\ \textrm {s}$ to $1.1\times 10^{-4}\ \textrm {s}$. For a given particle diameter, the particle volume fraction increases after the passage of the shock. The smaller particles respond faster than the larger ones. Thus, the gas–particle contact surface moves faster for the smaller particles.

Figure 8(b) shows the pressure evolution for different particle diameters. When the shock reaches the gas–particle contact surface, the pressure increases, generating two pressure waves in opposite directions: transmitted and reflected waves. The reflected wave is slower than the transmitted one. The attenuation effect of the small particles on the shock velocity is more evident. The velocity of the reflected pressure wave does not depend on the particle diameter.

The gas velocity evolution for the passage of a shock wave through a cloud of particles is depicted in figure 8(c). The small particles respond rapidly to the shock wave and the gas velocity is reduced immediately after the shock passage. The larger particles are more difficult to accelerate as a result of high inertia, as one can see in figure 8(d), showing the evolution of the particles after the passage of a shock wave.

From figure 8, the general conclusion that can be drawn is that the particles of larger diameter have a stronger attenuation effect on the transmitted shock wave as well as on the reflected pressure wave profiles. The opposite is true for the transmitted shock-wave velocity, i.e. the smaller the particles are, the slower the corresponding shock wave is. The interesting fact is that the velocity of the reflected pressure wave does not depend on the particle diameter. Moreover, analyses of the computed results show that the entropy measure, i.e. $P/\rho ^\gamma$, is nearly constant across the reflected wave. This property is used in the reduced-order modelling.

5.6. Effects of the particle volume fraction

In this section, the effects of the initial particle volume fraction $\tau _{v,0}$ are investigated. Particles of diameter $10\ \mathrm {\mu } \textrm {m}$ are used with volume fractions varying from $5\times 10^{-5}$ to $5\times 10^{-3}$.

Figure 9(a) gives the pressure evolutions after the shock passage. The particles of volume fraction $\tau _{v,0} = 5\times 10^{-5}$ have a very small influence on the pressure evolution. A high volume fraction $\tau _{v,0} = 5\times 10^{-3}$ seems to totally attenuate the transmitted shock at around $x = 0.7\ \textrm {m}$. The reflected pressure wave can be noted for all particle volume fractions. The comparison shows that a high particle volume fraction can increase the reflected pressure magnitude and attenuate the transmitted shock wave. Moreover, the reflected shock waves have similar velocities for different volume fractions. The gas velocity evolutions for different particle volume fractions are presented in figure 9(b). The reduction of the gas velocity is much reinforced by the increase of the particle volume fraction.

Figure 9.

Evolution with distance of normalized (a) pressure $p$ and (b) gas velocity $u_g$, at $t=600\ \mathrm {\mu } \textrm {s}$, $M_s=1.1$, $D = 10\ \mathrm {\mu } \textrm {m}$, $p_0 = 1.013\ \textrm {bar}$ and $u_{g,0}=55.19\ \textrm {m}\,\textrm {s}^{-1}$, for different particle volume fractions. Curves: $\tau _{v,0} = 5\times 10^{-5}$ (dark blue); $\tau _{v,0} = 1\times 10^{-4}$ (blue-green); $\tau _{v,0} = 2\times 10^{-4}$ (blue); $\tau _{v,0} = 5\times 10^{-4}$ (green); $\tau _{v,0} = 1\times 10^{-3}$ (orange); $\tau _{v,0} = 2\times 10^{-3}$ (orange-red); $\tau _{v,0} = 5\times 10^{-3}$ (red); and original gas–particle contact surface (black dashed).

6.1. Formation of the compressed gas zone

As illustrated in figure 1(a), initially the gas–particle contact surface separates the pure gas (1) from the gas–particle zone (4). After the passage of the shock, the pure gas zone (1) interacts directly with the post-shock gas–particle zone (3), before the formation of the compressed gas zone (2), where no reflected pressure expansion exists (see figure 10a).

Figure 10.

Sketch of the shock and gas–particle contact surface during the particle acceleration period. (a) Global acceleration configuration. (b) Local acceleration configuration in the proximity of the contact surface in the interface-attached frame. The red zone denotes the creation of the compressed gas. Here SG = shocked gas, G = gas, and P = particles.

In order to simplify the analysis, one can locate the origin of the coordinates at the gas–particle contact surface as illustrated in figure 10(b). Denoting the velocity of the contact surface as $V_i$, the gas velocity of the upstream flow is ($u_{g,1} - V_i$). Similarly, the post-shock gas velocity in zone (3) can be expressed as (${u}_{g,3} - V_i$). Mass conservation across the contact surface gives

(6.1)\begin{equation} \rho_{g,1}(u_{g,1} - V_i) = {\rho}_{g,3}( {u}_{g,3} - V_i). \end{equation}

Assuming that the gas velocity before the first particle $u_g$ is constant, we have the interface velocity $V_i(t) = u_{g,1}[1 - \exp (-t/\tau _p )]$, where $\tau _p$ is the particle response time.

The gas velocity in zone (3) can be estimated from the momentum conservation equation at point $x_0$:

(6.2)\begin{equation} \frac{\textrm{d}u(x_0,t)}{\textrm{d}t} = -\frac{m_p}{\rho_g\mathcal{V}_g} \frac{\textrm{d}V_i}{\textrm{d}t} = -\tau_{v,0}\frac{\rho_p}{\rho_g} \frac{\textrm{d}V_i}{\textrm{d}t}. \end{equation}

For the simplicity of the analysis, we assume that the particle response time with nonlinear correction term defined in (5.15a,b) remains constant during the particle acceleration process. Thus, one has

(6.3)\begin{equation} \frac{\textrm{d}V_i}{\textrm{d}t}=\frac{1}{\tau_p}(u(x_0,t) - V_i(x_0,t)). \end{equation}

The gas velocity before the first particle is assumed to be constant, $u(x_0,t)=u_{g,1}$. Considering that the gas properties around the contact particles are similar to those in the upstream flow $\rho _g = \rho _{g,1}$, one has

(6.4)\begin{equation} \frac{\textrm{d}u}{\textrm{d}t} = -\frac{\tau_{v,0}}{\tau_p}\frac{\rho_p}{\rho_{g,1}} u_{g,1}\exp\left(-\frac{t}{\tau_p}\right). \end{equation}

Knowing that $u = u_{g,1}$ at $t = 0$, the gas velocity of zone (3) can be obtained:

(6.5)\begin{equation} { u}_{g,3}(t) = u_{g,1}+\frac{\tau_{v,0}\rho_p}{\rho_{g,1}} u_{g,1}\left[\exp\left(-\frac{t}{\tau_p}\right)-1\right]. \end{equation}

By combining (6.5) and (6.1), one has

(6.6)\begin{equation} \frac{{\rho}_{g,3}(t)}{\rho_{g,1}} = \mathcal{A}_{\rho} =\frac{1}{1+\dfrac{\tau_{v,0}\rho_p}{\rho_{g,1}} -\dfrac{\tau_{v,0}\rho_p}{\rho_{g,1}}\exp\left(\dfrac{t}{\tau_p}\right)}. \end{equation}

The right-hand side of (6.6) can be seen as an amplification factor $\mathcal {A}_{\rho }$ of the gas density due to the deceleration of the gas flow in the presence of particles. This factor is plotted as a function of the normalized time in figure 11.

Figure 11.

Amplification factor $\mathcal {A}_{\rho }$ for two different incident Mach numbers: (a) $M_s = 1.1$, $\tau _p = 0.1\ \textrm {ms}$ with $D=10\ \mathrm {\mu } \textrm {m}$ and $\tau _{v,0} = 5.2\times 10^{-4}$; and (b) $M_s = 4.0$, $\tau _p = 21\ \mathrm {\mu } \textrm {s}$ with $D=10\ \mathrm {\mu } \textrm {m}$ and $\tau _{v,0} = 5.2\times 10^{-4}$.

With the assumption that the upstream gas density $\rho _{g,1}$ and velocity $u_{g,1}$ are constant during the acceleration process and to ensure the conservation of mass across the gas–particle contact surface, the density of the gas might diverge to a very high value, as well as the gas pressure inside the gas–particle zone (3) (see figure 11). This is the reason why a compressed gas in zone (2) is created aside from the gas–particle contact surface in the pure gas region (1).

6.2. Necessary condition for the density peak

Before the formation of the compressed gas zone, the post-shock gas density can increase up to a very high value at $t_{c}$ when

(6.7)\begin{equation} 1+\frac{\tau_{v,0}\rho_p}{\rho_{g,1}} - \frac{\tau_{v,0}\rho_p}{\rho_{g,1}}\exp\left(\frac{t_c}{\tau_p}\right) = 0. \end{equation}

One can see that the gas velocity in the gas–particle zone (3) decreases from $u_{g,1}$ to a lower value during the time interval $[0,t_{c}]$. A negative gas velocity gradient leads to a negative particle velocity gradient, which forms the number-density peak of particles inside the gas–particle zone (3), if this negative gradient exists for a long enough time in the gas–particle mixture.

However, if one has $t_{c}\approx O(\tau _p)$, the compressed gas zone is created immediately when the shock reaches the gas–particle contact surface. In this case, the particle number-density peak cannot be obtained. For example, when $D = 10\ \mathrm {\mu } \textrm {m}$, $M_s=1.1$, no number-density peak can be seen in figure 11(a). The condition $t_{c} \gg \tau _p$ seems to be necessary for the appearance of the number-density peak, since only in this case can the negative velocity gradient be obtained for a long enough period inside the gas–particle zone (3).

The amplitude of the number-density peak is related to two factors: the residence time of the negative gas velocity gradient $t_{c}$, and the amplitude of the density change, which can be determined by both the particle cloud and the shock-wave intensity. In other words, this phenomenon is associated with the deceleration capacity of the particles characterized by $\tau _p$ and $\tau _{v,0}$ as well as the incident Mach number $M_s$.

The prediction of the number-density peak is confirmed by the numerical simulations and presented in figure 12(a). For a given volume fraction $\tau _{v,0} = 5.2\times 10^{-4}$, particles of diameter $D = 10\ \mathrm {\mu } \textrm {m}$ give a number-density peak after the passage of a shock wave of Mach number $M_s = 4.0$ ($t_{c} \gg \tau _p$). It is seen that the density peak increases with the initial Mach number $M_s$. The particle number-density peak is depicted in figure 12(a); it differs from the volume fraction ramp presented in Saito, Marumoto & Takayama (Reference Saito, Marumoto and Takayama2003). The number density is located inside the ramp and has a higher value for particle volume fraction.

Figure 12.

Evolution of (a) particle volume fraction $\tau _v$, (b) gas mass density $\rho _g$, (c) gas velocity $u_g$ and (d) particle mean velocity $v_p$, at $t=300\ \mathrm {\mu } \textrm {s}$, $D = 10\ \mathrm {\mu } \textrm {m}$ and $\tau _{v,0} = 5.2\times 10^{-4}$, for different Mach numbers. Curves: $M_s = 1.1$ (dark blue long dashed); $M_s = 1.5$ (blue-green dashed); $M_s = 2.0$ (blue dot-dashed); $M_s = 2.5$ (green long/short dashed); $M_s = 3.0$ (yellow dotted); $M_s = 4.0$ (red solid); and original gas–particle contact surface (black dashed).

The evolution of the gas density is shown in figure 12(b). One can note that the gas density increases abruptly at the location of the particle number-density peak. The decrease of the gas density downstream of the transmitted shock is due to the two-way coupling. The evolutions of the gas and particle mean velocity are given in figure 12(c) and (d), respectively.