2.1. Free interaction theory and the minimum entropy production principle

The free interaction theory is proposed firstly by Chapman et al. (Reference Chapman, Kuehn and Larson1958), according to which the pressure rise $p_{\textit{FIT}}/p_\infty$ in the separation region as shown in figure 2(b) can be expressed as follows:

(2.1)\begin{equation} \frac{p_{\textit{FIT}}}{p_\infty}=1+F(\bar{x}){\gamma}{M_\infty}^2\sqrt{\frac{C_{f0}}{2({M_\infty}^2-1)^{0.5}}}, \end{equation}

herein $\gamma$ is the specific heat ratio and $F(\bar {x})$ is a universal correlation function which depends on pressure rise distance and flow patterns. In turbulent flow, $F(\bar {x})$ is proposed by Erdos & Pallone (Reference Erdos and Pallone1962) such that $F(\bar {x})_{tur0}\approx 4.22$ close to separation point and $F(\bar {x})_{tur1}\approx 6$ for pressure plateau, while in laminar flow, $F(\bar {x})$ is found experimentally such that $F(\bar {x})_{lam0}\approx 0.8$ and $F(\bar {x})_{lam1}\approx 1.5$, respectively (see Hakkinen et al. Reference Hakkinen, Greber, Trilling and Abarbanel1959; Babinsky & Harvey Reference Babinsky and Harvey2011). Here, $C_{f0}$ is skin friction coefficient, which could be obtained according to incoming flow conditions (see Tao et al. Reference Tao, Fan and Zhao2014). Therefore, the independent variables of FIT are upstream flow properties including free stream Mach number $M_\infty$ and skin friction coefficient $C_{f0}$.

The minimum entropy production principle is employed firstly by Li & Ben-Dor (Reference Li and Ben-Dor1996a,Reference Li and Ben-Dorb) to determine the criteria of RR to MR transition in steady as well as unsteady shock–shock interaction flow, based on which the MEP is employed to analyse the separation shock–separation shock interactions by Wang et al. (Reference Wang, Xue and Cheng2018). In the flow shown in figure 4(a), an implicit function is derived to establish the connection among the configuration of separation shock–separation shock interaction, total entropy production and flow conditions, which is expressed as follows:

(2.2)\begin{equation} \ddot{S}_{RR}=f_{SRR}(M_\infty,\beta_i,\bar{\bar{p}}), \end{equation}

where total entropy production factor $\ddot {S}_{RR}$ is related to variables of the incoming Mach number $M_\infty$, separation shock angle $\beta _i$ and equivalent back pressure $\bar {\bar {p}}$. The implicit relation of (2.2) is determined by the expressions of $\ddot {S}_{RR}$ and $\bar {\bar {p}}$.

Here, $\ddot {S}_{RR}$ is derived by employing the incoming mass flow rate (two-dimensional flow) to normalize total entropy production $\dot {S}=\int \rho u{\rm \Delta} s \,\textrm {d}y$ (see Li & Ben-Dor Reference Li and Ben-Dor1996a; Wang et al. Reference Wang, Xue and Cheng2018)

(2.3)\begin{equation} \ddot{S}_{RR}=-\frac{\displaystyle\int\rho_rM_r\sqrt{T_r}\ln(p_{0r}/p_{0\infty})\,\textrm{d}y}{l\rho_\infty M_\infty \sqrt{T_\infty}}, \end{equation}

where $\rho$, $M$, $T$, $p_0$ and $l$ denote the local density, Mach number, static temperature, total pressure and the height of incoming flow, respectively. The subscripts $\infty$ and $r$ denote far-field flow and flow crossing downstream reflected shock waves, respectively.

Here $\bar {\bar {p}}$ is expressed as follows:

(2.4)\begin{equation} \bar{\bar{p}}=\frac{\displaystyle\int\rho_iM_i\sqrt{T_i}p_i\,\textrm{d}y}{l\rho_\infty M_\infty \sqrt{T_\infty}}, \end{equation}

where subscript $i$ denotes flow crossing upstream incident (or separation) shock waves. According to the work of Wang et al. (Reference Wang, Xue and Cheng2018), the influence of downstream pressure disturbance on shock–shock interaction could be measured by $\bar {\bar {p}}$. Because the variables can be obtained by incoming Mach number $M_\infty$ and local shock angle $\beta _i$, for shock $i_1$–shock $i_2$ interaction flow shown in figure 4(a), (2.3) and (2.4) can be written as

(2.5)\begin{equation} \left.\begin{array}{@{}c@{}} \ddot{S}_{RR}=f(M_\infty,\beta_1,\beta_2),\\ \bar{\bar{p}}=g(M_\infty,\beta_1,\beta_2), \end{array} \right\} \end{equation}

herein $\beta _1$ and $\beta _2$ are shock angles of $i_1$ and $i_2$, respectively. Then (2.2) is derived by inserting $\beta _1=g^{-1}(M_\infty ,\beta _2,\bar {\bar {p}})$ into $f(M_\infty ,\beta _1,\beta _2)$. According to the MEP, for given flow conditions (neglect heat exchange), if $f_{SRR}(M_\infty ,\beta _2,\bar {\bar {p}})=\min$, (2.2) should fulfil the following limitations:

(2.6)\begin{equation} \left.\begin{array}{@{}c@{}} \displaystyle\dfrac{\partial f_{SRR}}{\partial \beta_2}=0,\\ \displaystyle\dfrac{\partial^2 f_{SRR}}{\partial {\beta_2}^2}\geq0, \end{array} \right\} \end{equation}

thus, for the flow fields shown in figure 4(a) with a given incoming Mach number, the correlation between separation shock strength and downstream pressure disturbance are determined, of which the theoretical solution path validated by experimental results is shown in figures 4(b) and 4(c).

Obviously, the two theoretical methods apply to interaction flow in different situations. On one hand, the FIT method attributes the separation shock strength only to boundary layer, indicating it could apply well to upstream portion of separation shock; on the other hand, the MEP method concentrates on the spatial structure of shock–shock interaction, which applies well to the downstream portion of separation shock with relatively large separation.

2.2. Proposed method for analysing regular SWBLI

Because the configurations of SWBLI might be affected by many factors including boundary layer (see Matheis & Hickel Reference Matheis and Hickel2015), incident shock strength (see Giepman et al. Reference Giepman, Schrijer and van Oudheusden2018) and downstream expansion fan (see Grossman & Bruce Reference Grossman and Bruce2018), the flow fields shown in figure 3(a) can be classified into such three flow components shown in figure5($a$–$c$): flow A of the wedge-induced flow, which is mainly determined by wedge angle and downstream expansion waves; flow B of the main interaction flow, which is related to incident shock and separation shock; flow C of the boundary layer flow and separation bubble, which can be affected by viscosity and skin friction. Here, B$_{AB}$ and B$_{BC}$ denote the boundaries between flow A and B and flow B and C, respectively. Streamlines in flow B (L$_B$) show weak effects of downstream flow, and reflected shock $r_1$ in this flow field is consequently straight and relatively strong, as shown in figure5($e$). However, in flow A, streamlines L$_A$ bend upward (or downward) because of expansion waves EW$_I$ (or EW$_S$), leading to a curved reflected shock $r_1$ in this flow field, as shown in figure 5($d$). It might be noticed that flow A is similar with inviscid flow, flow B can be assumed as an overall shock–shock interaction flow, and flow C mostly depends on boundary flow.

Figure 5.

Schematic illustration of classification for flows in SWBLI: ($a$) wedge induced flow; ($b$) main interaction flow; ($c$) boundary layer flow; ($d$) detailed flow A, reflected shock–expansion waves interaction; ($e$) detailed flow B, incident shock–separation shock interaction; and ($\,f$) detailed flow C, boundary layer-separation shock interaction.

The configuration of flow A is mainly characterized by shock–expansion waves interaction, as shown in figure 5($d$), in which the reflected shock $r_1$ could be computed in steps via assuming the expansion fan as a series of weak waves, e.g. the total flow deflection angle $\theta$ is replaced by finite small angles $\theta _n$ (the number is $\theta /\theta _n$). When each flow deflection angle $\theta _n$ is small enough, pressure rise $p_n/p_{n-1}$ and Mach number $M_n$ of flow crossing n waves could be obtained by the second-order approximate solution derived as follows:

(2.7)\begin{gather} \frac{p_n}{p_{n-1}}=1-\frac{\gamma {M_{n-1}}^2\theta_n}{\sqrt{{M_{n-1}}^2-1}}+\frac{\gamma {M_{n-1}}^2{\theta_n}^2[(\gamma+1){M_{n-1}}^4-4({M_{n-1}}^2-1)]}{4({M_{n-1}}^2-1)^2}, \end{gather}

(2.8)\begin{gather} M_n=\sqrt{\frac{2}{\gamma-1}\left[\left(1+\frac{\gamma-1}{2}{M_{n-1}}^2\right)\left(\frac{p_n}{p_{n-1}}\right)^{({1-\gamma})/{\gamma}}-1\right]}. \end{gather}

The local pressure rise $p'_n/p_n$, shock angle $\beta _n$ and flow deflection angle $\alpha _n$ of flow crossing $r_1$ can be expressed by following:

(2.9)\begin{gather} \frac{p'_n}{p_n}=\frac{2\gamma}{\gamma+1}{M_n}^2\sin^2\beta_n-\frac{\gamma-1}{\gamma+1}, \end{gather}

(2.10)\begin{gather} {M'_n}^2\sin^2(\beta_n-\alpha_n)=\frac{1+\dfrac{\gamma-1}{2}{M_n}^2\sin^2\beta_n}{\gamma {M_n}^2\sin^2\beta_n-\dfrac{\gamma-1}{2}}. \end{gather}

In addition, the pressure rise and flow deflection angles should be the same on both sides of slip line $s_n$

(2.11)\begin{equation} \left. \begin{array}{@{}c@{}} p'_n=p''_n,\\ \theta_n+\alpha_n=\alpha_{n-1}+\theta'_n, \end{array} \right\} \end{equation}

herein $p''_n$ could be obtained by (2.7) with $M'_{n-1}$ and $\theta '_n$, and $M'_{n-1}$ could be obtained by (2.10) with $M_{n-1}$, $\alpha _{n-1}$ and $\beta _{n-1}$. The equations are closed and then all the variables are determined in steps based on the incoming flow conditions.

Flow B is an incident shock–separation shock interaction. As shown in figure 5($e$), the main entropy is produced by all the incident and reflected shock waves, hence (2.3) could be written as

(2.12)\begin{equation} \ddot{S}_{RR}=-\frac{\displaystyle\int\rho_3M_3\sqrt{T_3}\ln(p_{03}/p_{0\infty})\,\textrm{d}y+\int\rho_4M_4\sqrt{T_4}\ln(p_{04}/p_{0\infty})\,\textrm{d}y}{l\rho_\infty M_\infty \sqrt{T_\infty}}.\end{equation}

Assume that all the shock waves $i_1$, $i_2$, $r_1$ and $r_2$ are straight and the local flow fields behind the shock waves are uniform, then (2.12) could be derived as

(2.13)\begin{align} \displaystyle\ddot{S}_{RR}&=-\frac{1}{\rho_\infty M_\infty \sqrt{T_\infty}} \left[k_{i1}\rho_1M_1\sqrt{T_1}\ln\left(\frac{p_{01}}{p_{0\infty}}\right)+ k_{i2}\rho_2M_2\sqrt{T_2}\ln\left(\frac{p_{02}}{p_{0\infty}}\right)\right.\nonumber\\ &\left.\quad +k_{r1}\rho_3M_3\sqrt{T_3}\ln\left(\frac{p_{03}}{p_{01}}\right)+ k_{r2}\rho_4M_4\sqrt{T_4}\ln\left(\frac{p_{04}}{p_{02}}\right)\right], \end{align}

herein the coefficients $k_{i1}$, $k_{i2}$, $k_{r1}$ and $k_{r2}$ are derived as follows:

(2.14)\begin{equation} \left.\begin{array}{@{}c@{}} \displaystyle k_{i1}=\dfrac{(\cos\beta_2+k_d\sin\beta_2)(\sin\beta_1-\alpha_1)}{\sin(\beta_1+\beta_2)},\\ \displaystyle k_{i2}=\dfrac{(\cos\beta_1-k_d\sin\beta_1)(\sin\beta_2-\alpha_2)}{\sin(\beta_1+\beta_2)},\\ \displaystyle k_{r1}=\dfrac{k_{i1}\sin(\beta_3-\alpha_1+\alpha_r)}{\sin\beta_3},\\ \displaystyle k_{r2}=\dfrac{k_{i2}\sin(\beta_4-\alpha_2-\alpha_r)}{\sin\beta_4}, \end{array} \right\} \end{equation}

where $k_d$ is a non-dimensional variable to denote the local position of the separation point, i.e. $k_d=d_s/h_i$, as shown in figure 5($e$). Here, $\alpha _r$ is the flow deflection angle on both sides of the slip line according to the solution of regular reflection with incident shock waves. Here, (2.4) should be written as

(2.15)\begin{equation} \bar{\bar{p}}=\frac{p_1\rho_1M_1\sqrt{T_1}k_{i1}+p_2\rho_2M_2\sqrt{T_2}k_{i2}}{\rho_\infty M_\infty \sqrt{T_\infty}}. \end{equation}

In addition, reflected shock angles, $\beta _3$ and $\beta _4$, are determined by incoming Mach number $M_\infty$ and incident shock angles $\beta _1$ and $\beta _2$. Thus $\beta _2$ depends on $M_\infty$, $\beta _1$ and $\bar {\bar {p}}$ under the limitations of (2.6). Then $\beta _2$ could be obtained from the implicit functions by iterative solution methods. Herein $\bar {\bar {p}}$ represents downstream disturbance exerted on interaction, of which the lower bound corresponds to ideal flow conditions without any disturbance (see Wang et al. Reference Wang, Xue and Cheng2018).

Flow C is boundary flow, in which the pressure rise $p_{\textit{FIT}}/p_\infty$ is obtained by (2.1) with given $M_\infty$ and $C_{f0}$ based on the FIT method. Then separation shock angle $\beta _{\textit{FIT}}$ ($\beta _{\textit {FIT}0}$ or $\beta _{\textit {FIT}1}$) could be computed by following:

(2.16)\begin{equation} \beta_{\textit{FIT}}=\arcsin\left[\left(\frac{p_{\textit{FIT}}}{p_\infty}+\frac{\gamma-1}{\gamma+1}\right) \frac{\gamma+1}{2\gamma{M_\infty}^2}\right]^{0.5}. \end{equation}

It should be noted that $M_\infty$ should be replaced by $M_{1n}$ inside of the boundary layer as shown in figure 5($\,f$), which means the separation shock bends downward because of the lower Mach number.

4.1. Influence of the MEP solution on SWBLI

A non-dimensional parameter $AR_{eff}$ defined by Grossman & Bruce (Reference Grossman and Bruce2018) is employed to measure the effect of the test section width divided by the height of the model upstream leading edge. The downstream expansion fan, which was observed by Grossman & Bruce (Reference Grossman and Bruce2018) shown in figure 3, could exert a strong influence on SWBLI with $AR_{eff}=1.0$. Therefore, in order to research the effect of the MEP solution on SWBLI, the models with various angles are mounted at $AR_{eff}=1.50$ in the Mach number $M_\infty =2.5$ tests to weaken the effect of downstream expansion fan, of which the incoming total pressure and total temperature are set as $P_0=3.0$ bar and $T_0=285$ K, respectively.

Because of the three-dimensional interference on both sides of the wedge accompanied by a series of Mach waves, both the incident and separation shock waves close to the interaction point are very thick, as shown in figure 11($a$), resulting in the uncertainty of shock angle measurement. In the similar work of Grossman & Bruce (Reference Grossman and Bruce2018), who employed surface oil flow visualization to show separation region topology on the wind tunnel floor, the most upstream separation line was observed approximately straight while the three-dimensional flow effect seemed to be stronger in relatively downstream region. Therefore, the relatively upstream flow in the current study could be assumed as two-dimensional, and the upstream edge of shock line in schlieren image is consequently used for shock angle measurement. Additionally, although the separation points experience different positions, the influence of thickness change of boundary layer on SWBLI is small enough to be neglected, according to Grossman & Bruce (Reference Grossman and Bruce2018). An algorithm program for schlieren image quantization based on grey level (see Xue, Wang & Cheng Reference Xue, Wang and Cheng2018) is employed to detect the shock angles, by which the time history lines of incident shock angle on region A, separation shock angle close to interaction point on region B and separation shock angle close to boundary layer on region C were detected and illustrated in figure 11($b$). In the following sections, the average value ($AVG$) of angles is used for interaction analyses on shock polar lines, and the error bars are characterized by standard deviation (STDEV ) such that $\mathit {Upper bound}=\mathit {AVG}+\mathit {STDEV}$ and $\mathit {Lower bound}=\mathit {AVG}-\mathit {STDEV}$.

Figure 11.

Example of experimental result on the $19^\circ$ wedge model in $M_\infty =2.5$ flow: ($a$) schlieren image and ($b$) measured shock angles by program algorithm.

Figure 12 gives the schlieren images of overall flow fields and local flow fields induced by $\alpha _1=9^\circ$, $12^\circ$, $13^\circ$, $15^\circ$, $17^\circ$, $19^\circ$, $20^\circ$ and $21^\circ$ shock generators, respectively, indicating a strong influence of incident shock strength on SWBLI. As is indicated in figure 12, there is no doubt that the separation shock is a curved shock due to the differentiation between separation shock angles measured on starting and ending portions, including angles close to the interaction point denoted as $\beta _{\textit {Exp}.\,\textit {int}.}$ shown in figure12($b_1$–$b_8$) and angles close to the boundary layer denoted as $\beta _{\textit {Exp}.\,bou.}$ shown in figure 12($c_1$–$c_8$). The interaction point of the $9^\circ$ wedge is inside of the boundary layer following a very curved separation shock bending downward, as shown in figure 12($c_1$). The interaction points of other wedges, as shown in figure 12($c_2$–$c_8$), are out of the boundary layer with similar separation shock angles of $\beta _{\textit {Exp.}\,\textit{bou}.}\approx 30.9^\circ$, which agrees well with the FIT solution of $\beta _{\textit {FIT}0}\approx 30.8^\circ$, meanwhile, in figure 12($b_2$–$b_8$), the separation shock angles close to the interaction point are distinctly different with $\beta _{\textit {Exp}.\,\textit {int}.}$ increasing from $30.7^\circ$ to $34.1^\circ$, which is close to the MEP solution of $\beta _{\textit {MEP}}$ ranging from $29.1^\circ$ to $33.8^\circ$ changed by the incident shock strength. For the $9^\circ$ wedge with $\beta _{\textit {MEP}}\ll \beta _{\textit {FIT}0}$, the interaction point is inside of the boundary layer followed by a relatively small separation bubble, corresponding to figure 9($c$); for the $12^\circ$ and $13^\circ$ wedges with $\beta _{\textit {MEP}}<\beta _{\textit {FIT}0}$, the separation shock angles, which are slightly decreased from a lower portion to a higher one, seem to be suppressed by interaction points, resulting in convex separation shocks corresponding to figure 9($b$); for the $15^\circ$ wedge with $\beta _{\textit {MEP}}\approx \beta _{\textit {FIT}0}$, the separation shock is straight, which corresponds to figure 9($a$); for the $17^\circ$ to $21^\circ$ wedges with $\beta _{\textit {MEP}}>\beta _{\textit {FIT}0}$, however, the separation shock angle is increased with a distinctly growing separation bubble, resulting in concave separation shocks, which correspond to figure 9($d$).

Figure 12.

Schlieren images of SWBLI at $M_\infty =2.5$, $AR_{eff}=1.50$: $(a_1)\text {--}(a_8)$ are overall flow fields of $\alpha _1=9^\circ$ to $\alpha _1=21^\circ$, respectively, $(b_1)\text {--}(b_8)$ are local flow fields close to interaction point of $\alpha _1=9^\circ$ to $\alpha _1=21^\circ$, respectively, and $(c_1)\text {--}(c_8)$ are local flow fields close to boundary layer of $\alpha _1=9^\circ$ to $\alpha _1=21^\circ$, respectively.

Similar patterns are observed from the tests conducted at $M_\infty =2$, $AR_{eff}=1.38$, on four wedges including $\alpha _1=9^\circ$, $12\,^\circ$, $13\,^\circ$ and $15\,^\circ$, as shown in figure 13. The FIT solutions under the current conditions are $\beta _{\textit {FIT}0}\approx 37.5^\circ$, while ideal MEP solutions for RR are $\beta _{{\textit{MEP}}9^\circ }=34.5^\circ$, $\beta _{\textit{MEP}12^\circ }=36.2^\circ$ and $\beta _{\textit{MEP}13^\circ }=36.8^\circ$ (flow transited to MR with $\alpha _1\geq 15^\circ$), respectively. The interaction point of the $9^\circ$ wedge with $\beta _{\textit {MEP}}\ll \beta _{\textit {FIT}0}$ is still inside of the boundary layer, leading to a small separation, as shown in figures13($a_1$) and 13($b_1$). The other wedges ($\alpha _1=12^\circ$ to $15^\circ$) demonstrate a stable separation shock angle close to the boundary layer with $\beta _{\textit {Exp}.\,\textit{bou}.}\approx 37.3^\circ$, while angles close to the interaction point are $\beta _{\textit {Exp}.\,\textit {int}.}=36.5^\circ$, $37.1^\circ$ and $44.6^\circ$, respectively, followed by a separation bubble growing gradually, as shown in figure 13($b_2$–$b_4$). In order to conduct a comprehensive analysis, the major angles of theoretical results and measured angles from schlieren images are summarized in figure 14, which gives the statistical results with time average and error bars obtained from more than 200 successive schlieren images for each case.

Figure 13.

Schlieren images of SWBLI at $M_\infty =2$, $AR_{eff}=1.38$: $(a_1)\text {--}(a_4)$ are overall flow fields of $\alpha _1=9^\circ$ to $\alpha _1=15^\circ$, respectively, and $(b_1)\text {--}(b_4)$ are local flow fields of $\alpha _1=9^\circ$ to $\alpha _1=15^\circ$, respectively.

Figure 14.

Statistics of shock angles for contrast between theoretical solutions (ideal conditions without disturbance) and experimental results (total pressure 4 bar, total temperature 285 K): ($a$) $M_\infty =2$, $AR_{eff}=1.38$ and ($b$) $M_\infty =2.5$, $AR_{eff}=1.50$.

As is indicated in figure 14, the uncertainties of $\beta _{\textit {Exp}.\,\textit {int}.}$ and interaction point height $H_{\textit {Exp}.\,\textit {int}.}$ are intensified with the increasing of incident shock strength, especially in MR configuration shown in figure 14($a$), which means a larger separation bubble is more unstable than a smaller one. When the interaction point is out of the boundary layer, i.e. except $\alpha _1=9^\circ$, $\beta _{\textit {Exp}.\,\textit{bou}.}$ agrees well with the FIT solution $\beta _{\textit {FIT}0}$, meanwhile $\beta _{\textit {Exp}.\,\textit {int}.}$ grows approximately along the line of MEP solution $\beta _{\textit {MEP}}$ with the increasing of $\alpha _1$, and it agrees well with $\beta _{\textit {MEP}}$ when $\beta _{\textit {MEP}}\geq \beta _{\textit {FIT}0}$ at $M_\infty =2.5$, shown in figure 14($b$). Additionally, with the growth of separation shock strength, a convex separation shock exists in $\beta _{\textit {MEP}}<\beta _{\textit {FIT}0}$, while a concave separation shock exists in $\beta _{\textit {MEP}}>\beta _{\textit {FIT}0}$, and when separation shock strength grows strong enough, the flow configuration would be transited to MR, which proves the flow patterns shown in figure 9($e$). The interaction solution of SWBLI is shown in figure 15 by shock polar lines. In figure15($b$), separation shock is very close to $\beta _{\textit {FIT}0}$ with $\alpha _1\leq 15^\circ$ because the interaction point is close to the boundary layer, however, separation shock strength grows distinctly with $\alpha _1>15^\circ$ due to the interaction point getting further away from the boundary layer. Both figures 14 and 15 demonstrate that separation shock strength is not a constant value but could be changed by incident shock strength, and the effect of MEP solution on SWBLI is characterized by suppressing the growth of separation shock strength as well as separation bubble size with $\beta _{\textit {MEP}}<\beta _{\textit {FIT}0}$ while intensifying the growth with $\beta _{\textit {MEP}}>\beta _{\textit {FIT}0}$. Additionally, although flow deflection angles of $19^\circ$, $20^\circ$ and $21^\circ$ have reached the detachment condition angle of $17.95^\circ$ for $M_\infty =2.5$, while the solutions shown in figure15($b$) are still residing in RR domain, indicating the RR to MR transition might be delayed by the interaction of incident shock and separation shock according to the MEP solution.

Figure 15.

Interaction solution path of SWBLI on shock polar lines changed by incident shock strength: ($a$) $M_\infty =2$, $AR_{eff}=1.38$ and ($b$) $M_\infty =2.5$, $AR_{eff}=1.50$.

4.2. Influence of the FIT solution on SWBLI

In order to research the effect of the FIT solution on SWBLI, the following tests are conducted on the $12^\circ$ wedge at $M_\infty =2$, $AR_{eff}=1.38$, $T_0=285\ \textrm {K}$ with various unit Reynolds numbers of free stream: $\mathit {Re}\times 10^{-7}\ \textrm {m}\approx 1.19$, 1.49, 1.79, 2.08, 2.38, 2.68 and 2.98. Figure 16 gives the evolution of SWBLI with schlieren images, indicating the flow configuration is not so sensitive to the change of Reynolds number. The separation shock is still a convex shock, because the MEP solution for the $12^\circ$ wedge is $\beta _{\textit{MEP}12^\circ }=36.2^\circ$, while the FIT solution $\beta _{\textit {FIT}0}$ decreases slightly from $37.9^\circ$ to $37.3^\circ$ with Reynolds number increasing from $1.19\times 10^7/\textrm {m}$ to $2.98\times 10^7/\textrm {m}$.

Figure 16.

Schlieren images of SWBLI at $M_\infty =2$, $AR_{eff}=1.38$: $(a_1)\text {--}(a_7)$ are overall flow fields of $P_0 (\textrm {bar})=2.0$ to 5.0, respectively, and $(b_1)\text {--}(b_7)$ are local flow fields of $\mathit {Re}\times 10^{-7}\ \textrm {m}\approx 1.19$, 1.49, 1.79, 2.08, 2.38, 2.68 and 2.98, respectively.

The major angles as well as the RR solution path on shock polar lines are summarized in figure 17. It could be observed from figure 17($a$) that $\beta _{\textit {Exp}.\,\textit {bou}.}$ decreases slightly, which agrees well with $\beta _{\textit {FIT}0}$, meanwhile the change of $\beta _{\textit {Exp}.\,\textit {int}.}$, which is a little larger than $\beta _{{MEP}12^\circ }$, is not so distinct. It should be noticed that the interaction point height $H_{\textit {Exp}.\,\textit {int}.}$ decreases almost linearly with the increasing of Reynolds number and decreasing of boundary layer thickness, meaning lower $\beta _{\textit {FIT}0}$ corresponds to smaller separation bubble size. In figure 17($b$), although the polar line of $\beta _{\textit {FIT}0}$ moves downward in a small range, the RR solution path is almost stable. Therefore, the influence of Reynolds number in the range from $1.19\times 10^7/\textrm {m}$ to $2.98\times 10^7/\textrm {m}$ on SWBLI is very weak compared with the influence of incident shock strength mentioned above.

Figure 17.

Statistics of shock angles and interaction solution path at $M_\infty =2$, $\alpha _1=12^\circ$, $AR_{eff}=1.38$: ($a$) contrast between theoretical solutions and experimental results, ($b$) RR solution path of SWBLI on shock polar lines changed by Reynolds number.

4.3. Influence of installation height of model on SWBLI

The downstream expansion fan might affect SWBLI with models mounted at a high position, which has been proved by the work of Grossman & Bruce (Reference Grossman and Bruce2018) on models mounted at $AR_{eff}=1$ with various thicknesses. Therefore, the effect of models mounted at a low position on SWBLI should be taken into consideration, and further tests are conducted on the $12^\circ$ and $13^\circ$ wedges in $M_\infty =2$, $P_0=4.0\ \textrm {bar}$, $T_0=285\ \textrm {K}$ flow with the installation height reduced gradually from $AR_{eff}=1.38$ till unstart, as shown in figure18. The flow fields induced by the $12^\circ$ wedge are shown in figure 18($a_1$–$a_4$) with $AR_{eff}$ increasing from 1.38 to 1.58, where the SWBLI configuration is observed relatively stable in starting flows of $AR_{eff}<1.58$. Similar patterns are observed on the $13^\circ$ wedge shown in figure 18($a_5$–$a_8$), while starting flows exist in $AR_{eff}<1.52$, which is lower than that in the $12^\circ$ wedge flow.

Figure 18.

Schlieren images of SWBLI at $M_\infty =2$: $(a_1)\text {--}(a_4)$ are overall flow fields of $\alpha _1=12^\circ$ with $AR_{eff}=1.38$, 1.55, 1.57 and 1.58, respectively, $(a_5)\text {--}(a_8)$ are overall flow fields of $\alpha _1=13^\circ$ with $AR_{eff}=1.38$, 1.55, 1.57 and 1.58, respectively and $(b_1)\text {--}(b_8)$ are local flow fields corresponding to $(a_1)\text {--}(a_8)$, respectively.

Figure 19 gives the major angles and solution path on shock polar lines, in which separation shock angles agree well with theoretical results in starting flow and the effect of $AR_{eff}$ on the RR solution path is not distinct, indicating the influence of reducing model height from $AR_{eff}=1.38$ till unstart on SWBLI is very weak before the appearance of unstart. Therefore, the RR configuration of SWBLI mostly depends on incident shock strength compared with model mounted position.

Figure 19.

Statistics of shock angles and interaction solution path at $M_\infty =2$, $\alpha _1=12^\circ$ and $13^\circ$: ($a$) contrast between theoretical solutions and experimental results and ($b$) solution path of SWBLI on shock polar lines changed by installation height of model.