2.1. Statement of the problem

According to previous investigations (Tan et al. Reference Tan, Sun and Huang2012; Li et al. Reference Li, Chang, Xu, Yu, Bao and Song2017; Huang et al. Reference Huang, Tan, Sun and Wang2018), the shock train behaviour is closely associated with the boundary layer condition. The disturbance due to the SWBLI makes the boundary layer thicken or separate intermittently along the tunnel, which leads to different shock train behaviours. Thus, the problem is essentially how a moving shock train behaves in the SWBLI region. To define the problem, consider the interaction in the schematic shown in figure 1. In practice, the flow field in the isolator is complex, involving perturbations induced from the incoming flow and combustion. Thus, it is difficult to establish a fundamental understanding of the interaction between the shock train and the SWBLI. To simplify this investigation, the movement of the shock train is considered to be driven by a throttle device that induces little perturbation and is widely used in the study of shock trains (Hutchins et al. Reference Hutchins, Akella, Clemens, Donbar and Gogineni2014; Klomparens et al. Reference Klomparens, Driscoll and Gamba2016; Huang et al. Reference Huang, Tan, Sun and Wang2018; Vanstone et al. Reference Vanstone, Hashemi, Lingren, Akella, Clemens, Donbar and Gogineni2018; Xiong et al. Reference Xiong, Fan, Wang and Tao2018; Hunt & Gamba Reference Hunt and Gamba2019).

Figure 1.

Schematic of the problem domain.

Actually, the SWBLI is three-dimensional (3-D), and its effects have been investigated widely (Bruce et al. Reference Bruce, Burton, Titchener and Babinsky2011; Burton & Babinsky Reference Burton and Babinsky2012; Clemens & Narayanaswamy Reference Clemens and Narayanaswamy2014; Grossman & Bruce Reference Grossman and Bruce2018). In the SWBLI region, a primarily reverse flow zone exists in the centre of the channel and two tiny corner reverse flow zones are located separately close to the sidewalls. The corner separation results in the formation of compression waves at the corner. These compression waves interact with the leading leg of the primary shock to constitute an intermediate zone. In the intermediate zone, compression waves from the corner separation lead to a more gradual deceleration and gentler adverse pressure gradient. Consequently, the boundary layer does not separate in this area (see figure 20 of Burton & Babinsky (Reference Burton and Babinsky2012)). Compared with the central separation, the corner separation has less impact. In addition, according to the findings of Wang et al. (Reference Wang, Sandham, Hu and Liu2015), the separation line is straight over the central part of the channel, while the reattachment line is curved, making the separation length in the streamwise direction near the centreline longer. For a large aspect ratio, the central interaction zone is relatively two-dimensional (2-D) and the size is increased; in contrast, the effects of the zones in the corner are weakened (see figure 8 of Wang et al. (Reference Wang, Sandham, Hu and Liu2015)). The unstable movements of the shock train observed by Tan et al. (Reference Tan, Sun and Huang2012) and Huang et al. (Reference Huang, Tan, Sun and Wang2018), of which the aspect ratios satisfy the condition mentioned by Wang et al. (Reference Wang, Sandham, Hu and Liu2015), can also confirm this. Furthermore, in the investigation by Xu et al. (Reference Xu, Chang, Zhou and Yu2015), the unsteady shock train motion in the SWBLI region was presented by a 2-D numerical simulation in the absence of corner and sidewall effects. Based on the analyses above, it is reasonable to infer that the unsteady behaviour of the shock train in such case can be triggered by a 2-D interaction alone, and the 3-D effects will be ignored in the following analysis.

2.2. Shock train modelling

The question here is how to investigate the dynamical mechanism and criterion of the unsteady shock train movement in the isolator. The key to answering this question is to construct an appropriate fluid model for the shock train. Currently, the widely used computational fluid dynamics model is nonlinear and of high order. It can help display the details of the flow structures but it is not convenient for analysing the mechanism and criterion of the unsteady motion. Therefore, the premise in the study of the unsteady mechanism is to set up a low-order model, which has an advantage of distinctly displaying the physical mechanism associated with stability. In this section, the problem is further reduced to one-dimensional.

Some analytical models have been proposed to analyse the dynamics of a normal shock (Culick & Rogers Reference Culick and Rogers1983; MacMartin Reference MacMartin2004; Bruce & Babinsky Reference Bruce and Babinsky2008). For a shock train, which is more complex, some studies have attempted to actively predict the shock train location using system identification (Hutchins et al. Reference Hutchins, Akella, Clemens, Donbar and Gogineni2014; Vanstone et al. Reference Vanstone, Hashemi, Lingren, Akella, Clemens, Donbar and Gogineni2018). The system identification requires no knowledge of the shock train physics; however, a simple physics-based model was then developed (Vanstone et al. Reference Vanstone, Lingren and Clemens2018). Xiong et al. (Reference Xiong, Fan, Wang and Tao2018) proposed a correlation with high precision between the amplitude of the shock train motion and the flow parameters. Unfortunately, these shock train models may be not suitable for analysing the dynamical properties of the shock train, as the coupling between the upstream and downstream flow is ignored. The air column between the first shock and the duct exit also plays a role in the transient behaviour (Sugiyama et al. Reference Sugiyama, Takeda, Zhang, Okuda and Yamagishi1988). When the first separation shock moves forward, the flow in the shock train region will be reconstructed transiently. This perturbation has an impact on the flow condition at the throat in the flap region, and then the flow at the throat affects the entire shock train, making a feedback system.

We begin by stating the governing equations for the shock train. A detailed derivation of these equations has been presented by Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2017), in which the suitability of the model was examined experimentally and numerically. The difference is that minor modifications were made to the model in this study to achieve a better fit. The model assumptions are as follows: (a) ideal gas and inviscid flow assumptions with a constant ratio of specific heat $\gamma$ and gas constant $R$, but the skin friction coefficient, which is associated with the SWBLI, is considered; (b) quasi-one-dimensional flow; (c) the unsteadiness phenomena in the incoming flow and shock train region are not considered; and (d) the heat release due to combustion is ignored. To simplify the modelling, the motion of the first separation shock is also assumed to represent the entire shock train behaviour. Due to the limited information obtained from the experiment, the approximate parameters of the experimental flow fields are obtained by 2-D numerical simulations and are used to assist the modelling. The free interaction theory, which states the pressure rise behind the separation shock (Chapman, Kuhen & Larson Reference Chapman, Kuhen and Larson1957), is introduced to relate the upstream and downstream conditions as follows:

(2.1)\begin{equation} \frac{p_2-p_1}{q_1} \propto \sqrt{\frac{2C_{f1}}{({M_1}^2-1)^{0.5}}}, \end{equation}

where $p$ is the wall pressure, $q$ is the dynamic pressure, ${C_f}$ is the skin friction coefficient, $M$ is the Mach number and the subscripts ‘1’ and ‘2’ denote the upstream and downstream conditions of the first separation shock, respectively. Because the wall pressure is highly nonlinear, we use ${0.5 \gamma p_{\infty }{M_1}^2}$ to approximate the dynamic pressure $q_1$. Considering that the Mach number in the current case is larger than 1.5, we approximate $({M_1}^2 - 1)^{-0.25}$ to ${{M_1}^{-0.5}}$ to simplify the calculation. On the other hand, for a moving shock at a velocity with respect to the channel, ${M_1}$ is replaced by $(u_1-\dot {x}_{{s}})/{a_1}$ (Culick & Rogers Reference Culick and Rogers1983), where $u$ is the velocity of flow, ${x}_{{s}}$ is the location of the STLE and $a$ is the speed of sound. Then we have

(2.2)\begin{equation} \frac{p_2-p_1}{p_{\infty}} = \frac{\gamma}{2}F\left(\frac{u_1-\dot{x}_{{s}}}{a_1}\right)^{1.5}\sqrt{2C_{f1}}, \end{equation}

where ${F}$ is a universal correlation function that is independent of Mach and Reynolds numbers (Chapman et al. Reference Chapman, Kuhen and Larson1957; Matheis & Hickel Reference Matheis and Hickel2015). It can be used to approximate the wall pressure evolution for a given Mach number and friction coefficient once ${F}$ has been determined by experiments or numerical simulations. For turbulent flow, a value of ${F} = 7.5$ produces a better agreement with the current experimental results (Li et al. Reference Li, Chang, Xu, Yu, Bao and Song2017).

A simple first-order transient response model (Cui, Wang & Yu Reference Cui, Wang and Yu2014) is used to describe the variation of the pressure behind the first separation shock ${p_2}$, which is given by

(2.3)\begin{equation} \dot{p}_2 = -\frac{p_2-{{\mathcal{P}}(x_{{s}}, p_{{b}})}}{\tau}, \end{equation}

where the subscript ‘$b$’ denotes the flow condition at the exit. The propagation time constant $\tau$ is calculated as $(x_{{b}}-x_{{s}})/a_2$ and the location of the throat $x_{{b}}$ is 0.46. Here, $\mathcal{P}(x_{{s}}, p_{{b}})$ is a function to obtain the pressure behind the first separation shock when the backpressure $p_{{b}}$ is disturbed (see Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2017), p. 8).

The coupling between the upstream and downstream flow, namely the duct volume effect, is described as follows:

(2.4)\begin{equation} \dot{m}_{\infty} - \dot{m}_{b} = \frac{\mathrm{d} {\rho_{b} V}}{\mathrm{d} t} = V\dot{\rho}_{b} + \rho_{b}\dot{V},\end{equation}

where $\dot {m}$ is the mass flow rate. The rate of the density variation at the exit $\dot {\rho }_{b}$ is calculated as $(\rho _{b}\dot {p}_{b}) / (\gamma p_{b})$ according to the equation of state, and the volume of the region downstream of the shock train $V$ is described as $A(x_{{b}}-x_{{s}})$ where $A$ is the duct area. The density of the separated flow is difficult to obtain in the modelling and the flow has a low velocity in this region, thus, we use the lumped parameter at the exit instead.

According to the equations above, the dynamic features of the shock train are solved based on its equilibrium states with different backpressures. In the shock train, the core flow is supersonic whereas the separation flow is subsonic. It is difficult to estimate the steady-state relationship between the upstream and downstream conditions. Thus, the Smart–Ortwerth model is employed. Heat release and loss are ignored, and then the simultaneous ordinary differential equations can be described as

(2.5a)\begin{gather} \frac{\mathrm{d} {M^2}}{\mathrm{d} (x/D_{{H}})} = -{M^2}\left(1+\frac{\gamma-1}{2}{M^2}\right) \left[\left(\frac{2}{{\gamma}p{M^2}}\frac{A}{A_{{c}}}\right)\frac{\mathrm{d} {p}}{\mathrm{d} (x/D_{{H}})}+ \left(4C_{f2}\frac{A}{A_{{c}}}\right)\right], \end{gather}

(2.5b)\begin{gather}\frac{\mathrm{d} ({{A_{{c}}}/A})}{\mathrm{d} (x/D_{{H}})} = \left[\frac{1-{M^2}(1-{\gamma}(1-{A_{{c}}}/A))}{{\gamma}p{M^2}}\right] \frac{\mathrm{d} {p}}{\mathrm{d} (x/D_{{H}})}+\left(\frac{1+({\gamma}-1){M^2}}{2}\right)4C_{f2}, \end{gather}

(2.5c)\begin{gather}\frac{\mathrm{d} {p}}{\mathrm{d} (x/D_{{H}})} = kC_{f\infty}\left(0.5{\gamma}p{M^2}\right), \end{gather}

where $A/A_c$ and $D_{{H}}$ are the normalized core flow area and hydraulic diameter, respectively. Here, $C_{f2}$, which represents the skin friction at the wall in the entire separated region, is assumed to be zero, $k$ is an empirical constant. According to a previous finding (see figure 8 of Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2017)), the pressure gradient in the shock train region described by (2.5c) is set as constant. The function $\mathcal{P}(x_{{s}}, p_{{b}})$ can be obtained by solving (2.5) with initial conditions, $p_1(x_{{s}})$, $M_1(x_{{s}})$, $A / A_{{c}}$ and terminal condition $p_{{b}}$.

Combined with the disturbance propagation model, duct volume effect and the equilibrium manifold linearization approach used by Cui et al. (Reference Cui, Wang and Yu2014), the equations describing the dynamics of the shock train motion can be concluded as follows:

(2.6a)\begin{gather} \dot{x}_{{s}} = - \frac{4\bar{a}_1(x_{{s}})}{3F{\gamma}p_{\infty}\sqrt{2\bar{M}_1(x_{{s}})\bar{C}_{f1}(x_{{s}})}} \left[p_2-\bar{p}_2(x_{{s}})\right], \end{gather}

(2.6b)\begin{gather}\dot{p}_2 = -\frac{\bar{a}_2(x_{{s}})}{x_{{b}}-x_{{s}}}\left[p_2-\mathcal{P}({x}_{{s}}, p_{{b}})\right], \end{gather}

(2.6c)\begin{gather}\dot{p}_{{b}} = -\frac{{\gamma}RT_{{t}}}{V\left[1+0.5({\gamma}-1){\bar{M}_{{b}}}^2\right]} \left[\dot{m}_{{b}}-\dot{m}_{\infty}\right]-\frac{{\gamma}{p}_{{b}}}{V}\dot{V}, \end{gather}

where ‘ $\bar { }$ ’ denotes the steady-state parameter. The speed of sound $\bar {a}_1$ and $\bar {a}_2$ are calculated with the static temperature of the incoming flow $T_{\infty }$ and the total temperature $T_{{t}}$, respectively. The distributions of $C_{f1}(x)$, $p_1(x)$ and the area-averaged Mach number $M_1(x)$ of the unthrottled flow are obtained in advance by a 2-D numerical simulation to assist the modelling, and the remaining parameters are solved by analytic expressions. Different from the previous work, the throttling ratio $A_{{b}} / A$ in the mass equation of (2.6c) is set as the input of the shock train system, and $p_{\infty }$ is used in the calculation instead of $p_1$ except for solving the ordinary differential equations. The validation of the shock train model is shown in appendix A.

2.3. Linearization of the shock train model

The dynamics equations (2.6) govern the evolution of the system state ${\boldsymbol{\mathsf{X}}} = [x_{{s}} \enspace p_2 \enspace p_{{b}}]^{\text {T}}$. Then, the equations can be written in the form

(2.7)\begin{equation} \dot{{\boldsymbol{\mathsf{X}}}} = \mathcal{J}({\boldsymbol{\mathsf{X}}}, t). \end{equation}

The linear stability analysis assumes the existence of an equilibrium solution ${\boldsymbol{\mathsf{X}}}_0$ for the system (2.6) referred to as the base flow and defined by $\mathcal{J}({\boldsymbol{\mathsf{X}}}_0) = 0$. Using the standard small perturbation technique, the instantaneous flow is decomposed into base flow and small disturbances ${\boldsymbol{\mathsf{X}}} = {\boldsymbol{\mathsf{X}}}_0 + \varepsilon {{\boldsymbol{\mathsf{X}}}}^\prime$, where $\varepsilon \ll 1$. The resulting equations are further simplified by considering that the perturbation is infinitesimal, allowing the nonlinear fluctuating terms to be neglected. Thus, the nonlinear dynamic equations (2.6) become a system of linear partial differential equations defined by

(2.8)\begin{equation} \dot{{\boldsymbol{\mathsf{X}}}}^\prime = {\boldsymbol{\mathsf{A}}}{{\boldsymbol{\mathsf{X}}}}^\prime, \end{equation}

where the vector ${{\boldsymbol{\mathsf{X}}}}^\prime =[{x_{{s}}}^\prime \enspace {p_2}^\prime \enspace {p_{{b}}}^\prime ]^{\text {T}}$ represents the perturbation variables. Here, ${\boldsymbol{\mathsf{A}}} = {{\mathrm {d} \mathcal{J}}/{\mathrm {d}{\boldsymbol{\mathsf{X}}}}}|_{{\boldsymbol{\mathsf{X}}}_0}$ is the Jacobian matrix obtained by linearizing the function $\textrm {Re}$ around the base flow ${\boldsymbol{\mathsf{X}}}_0$. To obtain the matrix ${\boldsymbol{\mathsf{A}}}$, we set ${{\boldsymbol{\mathsf{X}}}}^\prime = [{x_{{s}}}^\prime \enspace 0 \enspace 0]^{\text {T}}$ at ${\boldsymbol{\mathsf{X}}}_0$. Then we have

(2.9)\begin{equation} \dot{{\boldsymbol{\mathsf{X}}}}^\prime = {\boldsymbol{\mathsf{A}}}{{\boldsymbol{\mathsf{X}}}}^\prime = {\boldsymbol{\mathsf{A}}}\left[{x_{{s}}}^\prime \quad 0 \quad 0\right]^{\text{T}} = \frac{\mathcal{J}\left({{\boldsymbol{\mathsf{X}}}_0} + \varepsilon{{\boldsymbol{\mathsf{X}}}}^\prime\right) - \mathcal{J}\left({{\boldsymbol{\mathsf{X}}}_0}\right)}{\varepsilon}, \end{equation}

and the first column of the Jacobian matrix ${\boldsymbol{\mathsf{A}}}$ is given by

(2.10)\begin{equation} \left[a_{11} \quad a_{21} \quad a_{31}\right]^{\text{T}} = \frac{\mathcal{J}\left({{\boldsymbol{\mathsf{X}}}_0} + \varepsilon{{\boldsymbol{\mathsf{X}}}}^\prime\right) - \mathcal{J}\left({{\boldsymbol{\mathsf{X}}}_0}\right)}{\varepsilon{x_{{s}}}^\prime}. \end{equation}

Similarly, we obtain the other two columns by setting ${{\boldsymbol{\mathsf{X}}}}^\prime = [0 \enspace {p_2}^\prime \enspace 0]^{\text {T}}$ and ${{\boldsymbol{\mathsf{X}}}}^\prime = [0 \enspace 0 \enspace {p_{{b}}}^\prime ]^{\text {T}}$ at the same equilibrium point. Then, the eigenvalue of the linearized system $\lambda = \sigma + \omega {\textrm {i}}$ is also obtained at this equilibrium point. The sign of the leading eigenvalue's real part $\sigma$ determines whether the equilibrium solution is linearly stable or unstable, whereas its imaginary part $\omega$ characterizes the stationary or oscillatory nature of the associated eigenvector.

3.1. Characteristics of the unsteady shock train movement

We took the flow conditions in the study by Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2017) as examples to present the characteristics of the unsteady movement of the shock train. Tests were conducted at Mach 1.85 and 2.70 and the stagnation pressures were 0.215 and $0.462\ \textrm {MPa} \pm 0.5\,\%$, respectively. The stagnation temperature was $305 \pm 5$ K. The thickness of the boundary layer at the exit of the wind tunnel was approximately 3.9 mm. This wind tunnel had an isolator test section that measures $50\ \text {mm wide} \times 30\ \text {mm tall} \times 320$ mm long. Fused silica windows were placed in the test section sidewalls for an optical access that measures $30\ \text {mm tall} \times 260\ \text {mm long}$.

In figure 2, the modelled trajectories are compared with the experimental ones. At a lower Mach number, as shown in figure 2(a), the model can predict the trend of STLE movement in general. According to the trajectory obtained from schlieren images, the oscillation emerges when the STLE enters the SWBLI region. For a higher Mach number, as shown in figure 2(b), a rapid forward movement is observed instead of the oscillation. Both the locations and moments when the unsteady motions occur in the model results are close to the experimental ones. It should be noted that the oscillation and rapid forward movement in different conditions are also distinguished by the model. The $\lambda$-type shock train occurs at a lower Mach number and the pressure increases quickly in the front part. Accordingly, a larger slope was used in (2.5c). In contrast, for the case with a higher Mach number, the shock train has a $\chi$-type structure and the pressure rise is more gradual. Thus, a smaller slope was used in (2.5c) during the modelling.

Figure 2.

The STLE trajectories at the bottom wall: (a) at $M_{\infty } = 1.85$, the values of ${0.5\gamma pM^2}$ and ${kC_f}$ in (2.5c) are assumed to be 85.0 kPa and 0.35 during the modelling and (b) at $M_{\infty } = 2.70$, the values of ${0.5\gamma pM^2}$ and ${kC_f}$ in (2.5c) are assumed to be 111.4 kPa and 0.16. The location is normalized with the isolator length $L_{{iso}}$ and the time is normalized with $L_{{iso}} / u_{\infty }$. The data are selected from the experimental results of Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2017). The entire movements of the STLEs at $M_{\infty } = 1.85$ and $M_{\infty } = 2.70$ can be seen in the supplementary movies 1 and 2 available at https://doi.org/10.1017/jfm.2020.702.

Due to the assumptions in the modelling, this shock train model has some shortcomings, which are embodied in the following aspects. The coefficient $k$ in the model may be unsuitable for other situations, due to the strong dependence on the flow conditions. Setting the coefficient as a constant may not be suitable for all cases. A data set or fitting function of $k$ obtained according to different flow conditions may give better performance, but numerous experimental data are required to support this process. Due to the dimensionality reduction in the modelling and simplification of the flow conditions, the modelled unsteady process lasts for less time than the experimental one. By using lumped parameters, which were obtained from a 2-D numerical simulation, parts of the effects of the SWBLI may be distorted. There exists a coupling between the movements of the STLEs at the lower and upper walls, however, this coupling is not modelled here, which also contributes to the disagreement. Nevertheless, the model presents the dynamic characteristics of the shock train within the SWBLI region and is considered to perform well considering the difficulty of mechanism modelling.

3.2. Linear stability analysis of the shock train movement

The time-domain behaviour suggests that there exists an instability mechanism inside the STLE movement. For a more detailed analysis, we focused on the linear stability analysis of the shock train behaviour. Figure 3 illustrates the details of the STLE movements in the second region of figure 2(a) and first region of figure 2(b). At the onset of the interaction between the STLE and SWBLI, the STLE oscillates with an increasing amplitude, which can be seen both in the model and experimental results. At a lower Mach number, the oscillation lasts until the STLE passes through the SWBLI region, of which the duration is approximately 0.300 s; whereas a rapid forward movement occurs instead at a higher Mach number and the duration is approximately 0.024 s. After the STLE crosses the SWBLI region, the decreasing amplitude of the oscillation suggests a decay process. According to the linearized equation set, at each equilibrium point, the Jacobian matrix ${{\boldsymbol{\mathsf{A}}}}$ has one purely real eigenvalue and a pair of conjugate eigenvalues. This suggests that when the STLE is perturbed, it moves to a new position with a decaying oscillatory path as shown in the enlarged view of figure 3, which is consistent with the experimental results of Vanstone et al. (Reference Vanstone, Hashemi, Lingren, Akella, Clemens, Donbar and Gogineni2018).

Figure 3.

Distributions of the flow parameters at the bottom wall and details of the STLE motions within the SWBLI regions: (a) $M_{\infty } = 1.85$ and (b) $M_{\infty } = 2.70$. The inset shows an enlarged view of the oscillatory path for the STLE. The dashed line with an arrow indicates the trend of the shock train movement.

As analysed in a previous study (see figure 7 of Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2017)), the shock train can be treated as a closed-loop system, and the characteristic is indicated by its eigenvalues (poles of the closed-loop system). Figure 4 depicts the variations of the three eigenvalues during the unsteady movement as shown in figure 3(a). As the parameters vary along the isolator, the feature of the shock train system is also disparate. To characterize this evolution, the eigenvalues were plotted against the STLE location and the corresponding changes of the parameters are with reference to figure 3(a). When the downstream STLE moves into the SWBLI region, the $C_f$ ahead of it decreases sharply, which indicates that the resistance of near-wall flow to an adverse pressure gradient becomes weak. Meanwhile the real parts of the conjugate eigenvalues increase above zero and STLE instability occurs. The evolution of the conjugate eigenvalues also indicates the occurrence of a Hopf bifurcation (Sipp et al. Reference Sipp, Marquet, Meliga and Barbagallo2010; Sansica et al. Reference Sansica, Robinet, Alizard and Goncalves2018). When $C_f$ decreases further, the real parts of the conjugate eigenvalues separate at some positions and the imaginary parts disappear. This can explain the sudden change in the amplitude in figure 3(a) and the rapid movement in figure 3(b). Due to a significant increase in $C_f$ when the STLE moves out of the SWBLI region, the stable real eigenvalue and conjugate eigenvalues emerge again, and then the STLE moves to a new position with a decaying oscillatory path. However, in the absence of a quantitative relationship, it would be arbitrary to relate $C_f$ with instability.

Figure 4.

Variations of the eigenvalues during the unsteady shock train movement: (a) real part and (b) imaginary part.

According to this qualitative analysis, we can only roughly associate the evolutions of the eigenvalues with the distributions of the parameters in the boundary layer. The existence of positive eigenvalues indicates that the shock train is unstable in the SWBLI region and a positive feedback mechanism exists. When the shock train is disturbed in a stable region, the flow makes the direction of $\dot {x}_{{s}}$ opposite to that of the offset, pulling the STLE back to the original position, which is called negative feedback. In contrast, if the shock train is disturbed in an unstable region, the flow makes the direction of $\dot {x}_{{s}}$ the same as that of the offset, pushing the STLE away from the original position, which is called positive feedback. However, the question of what is the underlying physics of positive feedback in the shock train within the SWBLI region is still open, and what causes the rapid movement is also still unknown. To find out what triggers the positive feedback and what causes the rapid movement of the shock train, we performed a quantitative analysis between the boundary layer conditions and the STLE stability as presented in the succeeding section.

4.1. Criterion for the instability of the shock train movement

Many studies have shown that the wall pressure gradient is responsible for the unstable movement caused by the SWBLI in the isolator (Tan et al. Reference Tan, Sun and Huang2012; Huang et al. Reference Huang, Tan, Sun and Wang2018). This criterion can be suitable for practical applications (Li et al. Reference Li, Chang, Xu, Yu and Song2019), but it may not be rigorous in theory. In this study, we deduced the criterion for the stability of the shock train motion theoretically. At first, we focused on the motion of the first separation shock, and its dynamic process described by (2.6a) and (2.6b) can be expressed as several integral processes with a physical feedback as shown in figure 5. When the STLE stabilizes at $x_{{s}}$, the steady-state relationship between the upstream and downstream conditions of the first separation shock can be obtained according to (2.2), of which the velocity $\dot {x}_{{s}}$ is set as zero. If the downstream flow is disturbed, the pressure behind the first separation shock deviates from $\bar {p}_2(x_{{s}})$ and reaches $p_2^+$. This process is not completed instantaneously, but takes a period of time. We can obtain the varying rate of the pressure through (2.6b). By an integral process the actual value of $p_2$ can be obtained. The deviation induced by the disturbance requires a certain speed of the STLE, which is equal to $\dot {x}_{{s}}$, to maintain the pressure ratio relationship. When the STLE moves to a new position where the local flow field can sustain $p_2^+$, $\dot {x}_{{s}}$ turns to be zero and then the shock train stabilizes. This process indicates that the motion of the STLE is a feedback system, which is characterized by a self-balance or a dynamic imbalance.

Figure 5.

Feedback mechanism of the STLE.

To analyse the feedback mechanism of the STLE motion, we defined the deviation $\varepsilon$ as

(4.1)\begin{equation} {\varepsilon} = p_2^+-\bar{p}_2(x_{{s}}), \end{equation}

where $\varepsilon$ is also the perturbation that provides the driving force, $p_2^+$ is the disturbed pressure behind the separation shock. If $\varepsilon$ is equal to zero, the STLE will be maintained at a fixed position.

For a given acoustic wave ${p_2}^+$, we get

(4.2)\begin{equation} \dot{\varepsilon} = -\dot{\bar{p}}_2(x_{{s}}) = -\frac{\mathrm{d} \bar{p}_2(x_{{s}})}{\mathrm{d} x_{{s}}}\dot{x}_{{s}}. \end{equation}

Then, according to (2.6a), (4.2) can be written as

(4.3)\begin{gather} \dot{\varepsilon} = -\frac{\mathrm{d} \bar{p}_2(x_{{s}})}{\mathrm{d} x}\dot{x}_{{s}} = \frac{\mathrm{d} \bar{p}_2(x_{{s}})}{\mathrm{d} x}\frac{4\bar{a}_1(x_{{s}})}{3F{\gamma}p_{\infty} \sqrt{2\bar{M}_1(x_{{s}})\bar{C}_{f1}(x_{{s}})}}\varepsilon = \lambda\varepsilon, \end{gather}

(4.4)\begin{gather}\lambda = \frac{\mathrm{d} \bar{p}_2(x_{{s}})}{\mathrm{d} x}\frac{4\bar{a}_1(x_{{s}})}{3F{\gamma}p_{\infty}\sqrt{2\bar{M}_1(x_{{s}})\bar{C}_{f1}(x_{{s}})}}, \end{gather}

where $\lambda$ is the eigenvalue of the deviation equation (4.1) for determining the stability of the self-feedback system. As known, the system will be unstable when $\lambda$ is positive and stable when $\lambda$ is negative. If $\lambda$ is equal to zero, the system will be at a critical state. As shown in (4.4), the sign of $\lambda$ is determined by ${\mathrm {d} \bar {p}_2(x_{{s}})/\mathrm {d} x}$. The term $\bar {p}_2(x_{{s}})$, which is calculated according to the free-interaction theory, indicates how much pressure rise the boundary layer can sustain in a supersonic flow, and it can also be considered as the critical pressure that stabilizes the shock.

Figure 6 presents the distributions of the wall pressure and $\bar {p}_2(x_{{s}})$ at the isolator bottom with the conditions involved in the study by Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2018). In the SWBLI region, the boundary layer undergoes a significant adverse pressure gradient, and with the wall pressure increasing, the boundary layer can sustain a larger pressure rise. This phenomenon is consistent with the findings of Do et al. (Reference Do, Im, Mungal and Cappelli2011a) and Im et al. (Reference Im, Baccarella, McGann, Wermer and Do2016), in which shock anchoring was observed in the region with a higher pressure gradient. It can also be confirmed in previous research (see figure 9 of Li et al. (Reference Li, Chang, Xu, Yu and Song2019)), where the SWBLI had a retardation effect on the STLE at the beginning of the period when it passed through. As shown in figure 6, when the STLE moves upstream across the peak A, the change of sign in the pressure gradient of $\bar {p}_2(x_{{s}})$ leads to a formation of the positive feedback mechanism. In the adverse pressure gradient region, the maximal pressure that the boundary layer can sustain continues to decrease with the STLE moving upstream. An offset in the upstream direction results in the same direction of the shock train movement, which is described by (2.6a), pushing the STLE away from the original position. Meanwhile, the bifurcation appears, transforming the system from stable to unstable. With an increasing backpressure, the STLE moves into the region with a favourable pressure gradient and stabilizes further upstream of the favourable pressure gradient region. The appearance of the favourable pressure gradient increases the maximal pressure that the boundary layer can sustain. An offset of the STLE position in the upstream direction results in a positive $\dot {x}_{{s}}$ (downstream direction) in (2.6a), which means that the negative feedback process pulls the STLE back to the onset. This provides the evidence that an adverse pressure gradient of $\bar {p}_2(x_{{s}})$, and not the wall pressure, is responsible for the Hopf bifurcation, and the favourable pressure gradient can increase the stability of the shock train.

Figure 6.

Distributions of wall pressure and maximal pressure $\bar {p}_2(x_{{s}})$ that the boundary layer can sustain in (a) case B in Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2018) and (b) case C in Li et al. (Reference Li, Chang, Xu, Yu, Bao and Song2018). The incoming Mach number of these two cases is 2.70, which is the same as that in figure 2(b). In case B, an additional angle of the wedge, which is used to generate the incident shocks, is set to achieve a different distribution in the SWBLI regions from those in figure 2(b). In case C, the distribution in the SWBLI regions is the same as that in figure 2(b), while the backpressure rising time is different. (Experiment, EXP; computational fluid dynamics, CFD.)

4.2. Mechanism of the rapid movement induced by SWBLI

Based on the analysis above, when the STLE passes through the SWBLI region, the Hopf bifurcation appears. The evolution of the limit cycle in the ${p_2}$–${x_s}$ plane is displayed in figure 7. When the STLE moves into the SWBLI region, the limit cycle grows from the equilibrium point and then remains stable. With the STLE moving forward, the limit cycle disappears and then the STLE stabilizes at a new equilibrium point. However, in figure 3, the STLE exhibits two different types of unstable behaviours: oscillation and rapid forward movement. In addition, when the shock train passes through point A in figure 6(a), it behaves more like an oscillation than a rapid movement, although the incoming flow conditions are similar with those in figure 6(b). Then, this question arises: Why are there different unstable motions? The Hopf bifurcation is usually accompanied by a limit cycle oscillation. Thus, we focused on the limit cycle in the following, that is: Under what condition is the trajectory closed?

Figure 7.

Evolution of the limit cycle during the STLE unstable movement based on the case in figure 3(a): (a) model result and (b) experimental result. The pressure ${p_2}$ in the experimental result is obtained by interpolation with the measured pressure distribution and STLE position. Due to the effect of noise, a lowpass filter with a passband frequency of 35 Hz is adopted.

For a fixed backpressure, the system (2.6) can be reduced into (4.5b) as follows:

(4.5a)\begin{gather} \dot{x}_{{s}} = f(x_{{s}}, p_2) = - \frac{4\bar{a}_1(x_{{s}})}{3F{\gamma}p_{\infty} \sqrt{2\bar{M}_1(x_{{s}})\bar{C}_{f1}(x_{{s}})}}\left[p_2-\bar{p}_2(x_{{s}})\right], \end{gather}

(4.5b)\begin{gather}\dot{p}_2 = g(x_{{s}}, p_2) = -\frac{\bar{a}_2(x_{{s}})}{x_{{b}}-x_{{s}}}\left[p_2-\mathcal{P}({x}_{{s}}, p_{{b}})\right]. \end{gather}

Not much is known about the conditions for the existence of the limit cycle; therefore, we focused our attention on the conditions for its non-existence. If the sign of ${\mathrm {d} f(x_{{s}}, p_2)}/{\mathrm {d} x_{{s}}}+{\mathrm {d} g(x_{{s}}, p_2)}/{\mathrm {d} p_2}$ does not change in the domain region, there is no closed trajectory for the system. We have

(4.6)\begin{equation} \frac{\mathrm{d} f(x_{{s}}, p_2)}{\mathrm{d} x_{{s}}}+\frac{\mathrm{d} g(x_{{s}}, p_2)}{\mathrm{d} p_2} = \frac{\dot{x}_{{s}}}{G}\frac{\mathrm{d} G}{\mathrm{d} x_{{s}}}-{G}\frac{\mathrm{d} \bar{p}_2(x_{{s}})}{\mathrm{d} x_{{s}}}+H, \end{equation}

where $G=- {4\bar {a}_1(x_{{s}})}/({3F{\gamma }p_{\infty }\sqrt {2\bar {M}_1(x_{{s}})\bar {C}_{f1}(x_{{s}})}})$ and $H=-{\bar {a}_2(x_{{s}})}/({x_{{b}}-x_{{s}}})$.

The divergence of the system can be obtained through a numerical solution as shown in figure 8(a). After the STLE crosses the critical point, the divergence increases rapidly first and then drops sharply through the zero axis. Meanwhile, the STLE oscillates to divergence and then a rapid movement, which is caused by the unstable real eigenvalue, occurs. According to the numerical solution in figure 8(a), after the rapid movement, there exists a region upstream of the STLE in which a sign reversal of (4.6) can occur. Based on the expression in (4.6), the divergence of system (4.5) can be influenced by the parameter gradients in the boundary layer. In addition, in figure 6, the limit cycle tends to exist in the situation with a gentler gradient (refer to the region boxed by the dotted line). For a further verification, the distributions of $C_{f}(x)$, $p(x)$ and $M(x)$ were modified based on the case in figure 3(a) to construct a new situation in which after the rapid movement, the sign of (4.6) will not change as shown in figure 8(b). The region with an adverse gradient was shortened with a dimensionless length of 0.020 as shown in figure 9(a) to obtain a steep gradient. Compared with the original trajectory shown in figure 3(a), no limit cycle is observed which is similar to the case in figure 3(b).

Figure 8.

Variation of the divergence of the system (4.1): (a) original case in figure 3(a), (b) modified case.

Figure 9.

(a) wall pressure distribution in the case of figure 8(b), (b) corresponding STLE trajectory.

For now, it can be concluded that the unsteady movement is induced by the intrinsic dynamics of the shock train and its stability is affected by the gradient of the maximal pressure ($\bar {p}_2(x_{{s}})$) that the boundary layer can sustain. Steeper adverse pressure gradients may result in a rapid movement of the shock train, while gentler adverse pressure gradients tend to induce oscillations. For the case involving a rapid movement, the significant pressure rise in $\bar {p}_2(x_{{s}})$ can be treated as a ‘dam’. When the STLE crosses the critical point, the shock train will release what it has been suppressing even with a small smooth change in the backpressure. At the critical point, the pressure that the boundary layer can sustain is relatively high and the forepart of the shock train has already sustained a certain pressure; therefore, it reaches the backpressure by extending only a relatively short distance. With a slight variation in the backpressure, the STLE enters into the region where the ability of the boundary layer to suppress the pressure is greatly reduced. Because the pressure downstream of the first separation shock is relatively low, the shock train has to extend a longer distance to reach the similar backpressure. This phenomenon is similar to the case that involves bleeding or ejecting (He, Huang & Yu Reference He, Huang and Yu2016; He et al. Reference He, Chang, Bao, Huang and Yu2016). A flow control can improve the flow state in the boundary layer and enhance the maximal pressure that the boundary layer can sustain. However, when the backpressure ratio is increased above the critical value, the shock train suddenly relocates upstream. For a gentler adverse pressure gradient, after this extension the initial movement continues due to the negative damping, and attains a limiting value due to the control of the negative damping by the increased positive damping, which is associated with the local flow condition. The positive damping becomes predominant and further reduces the amplitude of the movement to a certain value and again the negative damping becomes predominant and continues to increase the amplitude. Then the limit cycle oscillation emerges. For a steeper adverse pressure gradient, the positive damping dominates alone and reduces the amplitude to a certain minimum value, during which only a rapid movement is observed. This finding provides a possible method of improving the stability of the shock train, namely, eliminating the adverse pressure gradient region, or at least, turning the rapid movement into oscillation by moderating the adverse pressure gradient when the adverse pressure gradient is inevitable.