2.1. The FIT

Classic FIT shows that once the separation shock is strong enough, the pressure rise produced by the separated region is governed solely by the local flow properties and is independent of the overall geometry (Chapman et al. Reference Chapman, Kuehn and Larson1958). Erdos & Pallone (Reference Erdos and Pallone1962) further argued that this pressure rise is controlled by the factor

(2.1) \begin{align} \frac {p-p_{0}}{q_{0}} = F(\overline {x})\,\sqrt {\frac {2C_{\kern-1pt f,0}}{M^{2}_0-1}}. \end{align}

Here, the subscript 0 denotes the interaction onset, $p$ is the pressure, which is normalised by the dynamic pressure $q_0=\gamma p_0 M^2_0 /2$ , $M$ is the freestream Mach number, and $C_{\kern-1pt f}$ is the skin-friction coefficient. The dimensionless function $F(\overline {x})$ is defined as

(2.2) \begin{align} F(\overline {x}) = \sqrt {f_1(\overline {x})\,f_2(\overline {x})}, \end{align}

where

(2.3) \begin{align} f_1(\overline {x}) = \int _{\overline {x}_0}^{\overline {x}}\left (\frac {\partial \overline {\tau }}{\partial \overline {y}}\right)_{w}{\rm d}\overline {x},\quad f_2(\overline {x}) = \frac {{\rm d}\overline {\delta ^*}}{{\rm d}\overline {x}},\quad \overline {\tau } = \frac {\tau }{\tau _{w,0}},\quad \overline {y} = \frac {y}{\delta ^*},\quad \overline {x} = \frac {x-x_0}{L_0}, \end{align}

with $\tau$ the shear stress, $\tau _{w,0}$ the wall shear stress at the interaction origin $x_0$ , $\delta ^*$ the boundary-layer displacement thickness, and $L_0$ a characteristic streamwise extent of the free-interaction zone. The dimensionless function $f_1(\overline {x})$ represents the streamwise integral of the dimensionless wall shear stress gradient from the onset of interference, characterising the accumulation of streamwise pressure gradients. And the dimensionless function $f_2(\overline {x})$ is the streamwise derivative of the dimensionless boundary-layer displacement thickness, used to characterise the flow deflection angle resulting from boundary-layer separation. Consequently, the independent variables of the FIT model are the upstream flow properties, namely the freestream Mach number and the local skin-friction coefficient $C_{\kern-1pt f,0}$ . The value of the universal pressure-rise function $F(\overline {x})$ at the reference station is fixed by FIT; it exhibits an initially steep rise followed by a more gradual increase that terminates at the plateau pressure of the nearly constant pressure separation bubble. For a turbulent boundary layer, $F(\overline {x})\approx 4.22$ at incipient separation, and it reaches 6.0 at the plateau pressure level within the separated region.

2.2. Proposed method for analysing SWBLI on the curved surface

Based on the relative position between the separation region and the onset of wall curvature, the SWBLI over a curved surface can be clearly classified into two types as illustrated in figure 1. In type-I interaction, the incident shock impinges on the convex curved surface near its onset, and separation initiates on the flat plate upstream of the curvature origin. In type-II interaction, the shock impinges further downstream on the convex surface, and separation occurs on the curved wall downstream of the curvature origin. For separated flows over curved walls, the classical FIT is no longer strictly valid: the boundary-layer equations derived for a flat plate incorrectly relate the pressure gradient to the wall shear when the local surface curvature is neglected. We therefore start from the two-dimensional boundary-layer equations written in a wall-based coordinate system:

(2.4) \begin{align} \rho \left ( \frac {\partial u}{\partial t} +\frac {u}{1+\kappa y}\frac {\partial u}{\partial x} +v\frac {\partial u}{\partial y} +\frac {\kappa }{1+\kappa y}uv \right ) = \frac {1}{1+\kappa y}\left (-\frac {\partial p}{\partial x} +\frac {\partial \sigma _x}{\partial x}\right ) +\frac {2\kappa \tau }{1+\kappa y} +\frac {\partial \tau }{\partial y}, \end{align}

where $\kappa$ is the wall curvature, $x$ is the tangential coordinate along the surface, $y$ is the normal coordinate pointing away from the surface, $u$ is the tangential coordinate component, $v$ is the wall-normal velocity component, $\tau$ is the shear stress, and $\sigma _x$ is the normal stress in the $x$ direction. Specific information and derivation can be found in Appendix A. Applying the wall boundary conditions gives the near-wall pressure-gradient relation

(2.5) \begin{align} \frac {\partial p}{\partial x} =\left (\frac {\partial \tau }{\partial y}\right)_{\!w} +2\kappa \tau _{w}, \end{align}

where compared with the flat-plate result, an extra term $2\kappa \tau _w$ appears.

Figure 1.

Schematic diagram of separated flow field of shock wave incident on convex curved surface, for (a) type-I flow fields and (b) type-II flow fields.

Figure 2 illustrates the influence of wall curvature on the near-wall flow dynamics within a separated flow field. A control-volume analysis of the curved flow is performed:

(2.6) \begin{align} [(p+\mathrm{d}p)\mathrm{d}y-p\mathrm{d}y] \cos \left (\frac {\kappa\mathrm{d}x}{2}\right ) ={}& (\tau + \mathrm{d}\tau )(1+\kappa\mathrm{d}y)\,\mathrm{d}x-\tau \mathrm{d}x \nonumber \\ &{}+ [\tau \mathrm{d}y + (\tau + \mathrm{d}\tau )\mathrm{d}y] \sin \left (\frac {\kappa\mathrm{d}x}{2} \right )\!. \end{align}

Neglecting higher-order terms, the result is identical to that obtained from the streamwise momentum equation analysis. During the rotation following the wall curvature, the combined vertical shear stresses generate a component in the $x$ direction. In addition, due to the curvilinear coordinate system, the same angular deflection corresponds to a larger arc length at larger radii, which further modifies the streamwise momentum balance. Together, these two effects introduce an additional curvature-related term in (2.5). This term originates from two physical mechanisms during the motion of fluid elements along curved streamlines: the additional shear component generated by the continuous turning of streamlines, and the inward momentum flux increment from the outer flow. Its magnitude depends on the product of local curvature and wall shear stress $\kappa \tau _w$ , which macroscopically manifests as the balancing of the required centripetal force.

Figure 2.

Schematic diagram of the surface boundary layer near wall microparticle analysis.

Figure 3.

Schematic diagrams of the simplified analysis for type-I flow: (a) flat-plate configuration, showing the baseline inviscid flow structure; (b) type-I interaction on a convex curved surface.

Taking the type-I flow field as an example, figure 3 demonstrates its difference from the flat-plate case. Within the separation bubble, the wall shear stress $\kappa \tau _w$ becomes negative, rendering this contribution negative in the streamwise direction as well. Notably, this force term directly arising from geometric curvature does not act to directly alter the external pressure gradient. Its physical essence lies in the coupled system formed by the mutual adaptation between the inviscid pressure drop induced by the downstream wall turning and the distributed momentum transport effects triggered by the upstream wall curvature within the viscous boundary layer. The curvature term $\kappa \tau _w$ acts as a continuously distributed source term along the streamwise direction; its cumulative effect in the flow direction, under the final steady state, can match the downstream pressure conditions. The pressure reduction at the reattachment point is transmitted upstream through this curvature effect term, consequently altering the distribution of the pressure plateau within the separation zone.

This physical mechanism, where a distributed viscous effect adapts to a concentrated inviscid condition, shares fundamental similarities with the process in classical SWBLI, where pressure redistributes and reaches equilibrium through viscous actions within the boundary layer. Therefore, the influence of convex curvature can be understood as modifying the way the viscous response adapts to external pressure conditions, ultimately shaping the new pressure distribution within the separated region.

From the foregoing analysis, we obtain the first curvature-modified pressure-response relation of the extended FIT:

(2.7) \begin{align} \frac {p-p_{0}}{q_{0}} =C_{\kern-1pt f,0}\frac {L_{0}}{\delta ^*_0}\int _{\overline {x}_0}^{\overline {x}}\left(\frac {\partial \overline {\tau }}{\partial \overline {y}}\right)_{w}\mathrm{d}\overline {x} +\frac {2}{q_0}\int _{x_{s}}^{x}\kappa \tau _{w}\,\mathrm{d}x. \end{align}

In this equation, $x_s$ and $x_r$ are used to represent the positions of the separation point and reattachment point. In type-I flow (figure 3), although separation occurs on the flat plate, the pressure distribution within the separated region is modified by the downstream curvature. The cumulative effect of the curvature term in (2.7) integrated from separation to reattachment results in a lower reattachment pressure compared to the flat-plate case. This reduction propagates throughout the separation zone through the elliptic nature of the separated flow, effectively scaling down the entire pressure-rise curve, much like applying a uniform attenuation factor.

Relative to the interaction onset, the wall deflection angle at the reattachment point is given by

(2.8) \begin{align} \theta _{r}=\int _{x_{s}}^{x_{r}}\kappa \,\mathrm{d}x. \end{align}

With $p_{\textit{rc}}$ denoting the reattachment pressure for type-I flow, and $p_{\textit{rp}}$ that for the flat-plate case, the resulting downstream pressure change $p'_{r}=p_{\textit{rc}}-p_{\textit{rp}}$ can be approximated as

(2.9) \begin{align} \frac {p'_{r}}{p_{2}} =-\frac {\gamma M_{2}^{2}}{\sqrt {M_{2}^{2}-1}}\int _{x_{s}}^{x_{r}}\kappa \,\mathrm{d}x, \end{align}

where $p_2$ and $M_2$ are the pressure and Mach number immediately behind the incident shock wave, which are detailed in Appendix B. This reflects that the deflection angle of the reattaching flow changes as a result of the variation in wall angle, thereby reducing the pressure rise.

Inger (Reference Inger1982), employing a nonasymptotic analysis, showed that for a given turbulent interaction pressure distribution, the wall shear within the upstream interaction region obeys

(2.10) \begin{align} \frac {\tau '_{w}}{\tau _{w,0}}=C_{\tau }\sqrt {\beta _{pm}}\,\overline {p}(\overline {x}), \end{align}

where, $\tau '_{w}$ denotes the wall shear stress disturbance induced by shock–boundary layer interaction, $C_{\tau }$ is model constant for skin friction, $\beta _{pm}$ is the Prandtl–Glauert compressibility factor, and $\overline {p}(\overline {x})$ is the normalised pressure variation distribution. In this expression, the coefficient depends only weakly on the Reynolds number. Consequently, for identical separation conditions, the wall shear downstream of incipient separation should follow a self-similar distribution. The approach of the shear stress to its asymptotic value is governed essentially by the overall pressure rise produced by the incident shock.

The governing equation describing the boundary-layer pressure response in the absence of wall curvature is

(2.11) \begin{align} \frac {p-p_{0}}{q_{0}} =C_{\kern-1pt f,0}\frac {L_{0}}{\delta ^*_0}\int _{\overline {x}_0}^{\overline {x}}\left(\frac {\partial \overline {\tau }}{\partial \overline {y}}\right)_{w}\mathrm{d}\overline {x}. \end{align}

For cases where the upstream flow conditions remain unchanged and disturbances only exist downstream, both FIT and the preceding analysis indicate that separation is determined solely by the flow parameters at the interaction onset, independent of downstream conditions. Consequently, there exists a physical consistency between (2.7), which describes the pressure variation, and (2.11). The corresponding terms on the right-hand side, when normalised, remain consistent, and (2.7) reduces exactly to (2.11) in the limit of vanishing curvature. Therefore, the effect of wall curvature can be attributed almost entirely to the second term on the right-hand side of (2.7). Integrating over the entire separation region and subtracting the two equations yields the difference in reattachment pressure rise between the separated flow on the curved surface and the flat-plate case:

(2.12) \begin{align} p'_r=\int _{x_{s}}^{x_{r}}2\kappa \tau _{w}\,\mathrm{d}x. \end{align}

The additional pressure perturbation induced by the wall curvature at reattachment feeds back into the entire pressure-rise process and is compatible with the reduced pressure gradient inside the separation bubble.

From the preceding analysis, this parameter $p'_r$ directly reflects the variation of the pressure distribution after separation. By using (2.9) to replace (2.12), the pressure variation can be conveniently evaluated based on the wall geometric conditions. For ease of use, after normalisation, it can be written as

(2.13) \begin{align} \Delta \overline {p}_r =-\frac {p_2}{p_0} \frac {M^2_2}{M^2_0} \frac {1}{\sqrt {M^2_2-1}} \int _{x_{s}}^{x_r}\kappa \,\mathrm{d}x, \end{align}

with the detailed derivation provided in Appendix B.

This parameter quantifies the downstream influence that is transmitted upstream to the separation onset through the curvature-modified momentum balance (2.5); the pressure variation inside the bubble is therefore directly linked to this quantity. Viscosity smooths the pressure field relative to an inviscid solution, so the wall pressure responds globally to any local perturbation. Consequently, even if only the short region immediately upstream of reattachment is curved, the pressure downstream of the separation point is altered throughout. The plateau pressure can thus be expressed in terms of the reattachment-pressure change given by (2.13).

Figure 4.

Schematic diagram of curved surface boundary-layer displacement delay.

Figure 5.

Schematic diagrams of the inviscid simplified analysis for type-II flow: (a) flat-plate configuration, showing the baseline inviscid flow structure; (b) type-II interaction on a convex curved surface.

We now examine the type-II flow field sketched in figure 1(b). Here, the boundary layer separates while already on the curved surface, so the entire separation bubble lies downstream of the flat/curved junction. Compared with type-I, the upstream boundary layer is also exposed to the curved wall, but the physical mechanism differs from the curvature effect acting behind the separation point. Recalling the meaning of the fundamental dimensionless function in FIT, we treat the wall plus the boundary-layer displacement thickness as an ‘equivalent wall’ whose shape is dictated by the growth rate of $\delta ^*$ . This idea allows us to quantify how upstream curvature influences the pressure rise that drives separation. As indicated in figure 4, the direction along which the upstream influence propagates is approximately the Mach wave (expansion fan) angle:

(2.14) \begin{align} \theta _s=\arcsin (1/M_{0}). \end{align}

Following the same procedure used to derive (2.13), we obtain the correction factor for the dimensionless pressure-rise increment caused by the reduced effective deflection angle due to the displacement delay:

(2.15) \begin{align} L_{\delta }=\delta ^*_0 \tan ^{-1}(\theta _s). \end{align}

As illustrated in figure 5, the flow direction is determined based on the displacement concept, i.e. the external inviscid flow follows the direction of the equivalent wall, which is defined as the solid wall shape augmented by the boundary-layer displacement thickness. For the flat-plate case, the deflection angle $\varphi _p$ can be readily obtained as

(2.16) \begin{align} \varphi _p = \arctan \left ( \frac {\mathrm{d}\delta ^*}{\mathrm{d}x} \right ) \approx \frac {\mathrm{d}\delta ^*}{\mathrm{d}x}. \end{align}

Similarly, using the linearised form of the Prandtl–Meyer relation, another pressure response equation can be derived after transformation:

(2.17) \begin{align} \frac {p-p_{0}}{q_{0}} = \frac {2}{\sqrt {M^{2}_0-1}} \frac {\delta ^*_0}{L} \frac {\mathrm{d}\overline {\delta ^*}}{\mathrm{d}\overline {x}}. \end{align}

Combining this equation with (2.11) yields the classical FIT.

For the type-II flow shown in figure 5(b), however, the influence of the aforementioned displacement delay must be taken into account. Recalling the fundamental mechanism of incipient separation, separated flow results from the interaction between the local boundary layer and the adjacent inviscid flow. While separation at the innermost layer occurs relative to the solid wall, its interaction with the outer inviscid flow is modulated by variations in the displacement thickness. Under the small-angle approximation, the rate of change of the displacement thickness upstream of separation in type-II flow, relative to the flat-plate case, can be estimated with (2.14) and (2.15) as

(2.18) \begin{align} \frac {{\rm d}\delta ^*}{{\rm d}x} \approx \kappa L_{\delta } \approx \kappa \delta ^*_0 M_0. \end{align}

This delay in displacement causes a mismatch between the angle of the equivalent wall and the actual solid wall in the initial part of separation, thereby counteracting part of the flow deflection angle, and leading to a reduction in the separation pressure rise. From (2.18), this angular variation $\Delta \varphi _c$ can be estimated as

(2.19) \begin{align} \Delta \varphi _c \approx \int _{x_{s}-L_{\delta }}^{x_{s}}\kappa \,\mathrm{d}x. \end{align}

Taking the flow parameters at the interaction onset as reference, the corresponding pressure change at this location can be obtained via the Prandtl–Meyer relation:

(2.20) \begin{align} \Delta p_s = -\frac {\gamma M^{2}_0}{\sqrt {M^{2}_0-1}} p_0\, \Delta \varphi _c. \end{align}

Since this equivalent delay weakens the effective deflection angle caused by displacement effects, the resulting change in dimensionless pressure rise can be approximated as

(2.21) \begin{align} \Delta \overline {p}_{\kern-1pt s} =-\frac {1}{\sqrt {M^{2}_0-1}}\int _{x_{s}-L_{\delta }}^{x_{s}}\kappa \,\mathrm{d}x. \end{align}

This parameter directly alters the effective wall angle, thereby modifying the growth rate of the displacement thickness, and consequently the pressure rise required for separation. As a result, the plateau pressure is further reduced. Type-I flow corresponds to $\Delta \overline {p}_{\kern-1pt s} = 0$ . If the separation point lies on the curved surface, then $\Delta \overline {p}_{\kern-1pt s} \lt 0$ and the flow becomes type-II.

It is worth noting that for type-II interactions, the local skin-friction coefficient $C_{\kern-1pt f,0}$ is not taken from the flat-plate reference case. Instead, it is evaluated at the actual interaction onset $x_0$ on the curved surface, using the local wall shear stress and the local edge dynamic pressure. Therefore, the effect of the upstream convex-wall development on the incoming boundary layer is retained through the local values of $C_{\kern-1pt f,0}$ , $M_0$ , and the associated boundary-layer parameters.

Separation on a curved wall is inevitably influenced by the incoming boundary layer; the kinetic energy loss induced by viscosity is embedded in the velocity profile. To account for the effect of this relative momentum deficit on the separation plateau pressure, Xie et al. (Reference Xie, Yang, Zeng, Liao, Ding, Zhang and Guo2021) scaled the plateau value of the free-interaction pressure-rise function $F$ by the incompressible shape factor $H_i$ of the oncoming boundary layer, yielding a scaled plateau function $F^*$ . Here, we adopt the same approach, and rescale the curved-wall plateau pressure-rise function with $H_i$ :

(2.22) \begin{align} F^*_c=F_c \big(H_{i}^{m}-1 \big)^{n}, \end{align}

where the subscript $c$ denotes the curved-wall flow, and the exponents $m$ and $n$ are chosen to achieve the best collapse of the data; typical values are $n\approx 0.5$ and $m\approx 1.2$ . With this scaling, results obtained under different incoming boundary-layer conditions collapse onto a single curve. The resulting variation of the dimensionless pressure-rise function then satisfies

(2.23) \begin{align} \frac {\Delta F^*_c}{F^*_{p}} =\alpha _{r}\,\Delta \overline {p}_{r} +\alpha _{s}\,\Delta \overline {p}_{\kern-1pt s}, \end{align}

where the subscript $p$ refers to the flat-plate baseline, and $\alpha _{r}$ and $\alpha _{s}$ are coefficients to be determined. This relation identifies the key factors that alter the plateau pressure-rise function when wall curvature is present. In contrast to the classical free-interaction conclusion, we augment the set of independent variables for curved-wall separation by introducing both an upstream-flow-shape parameter and a downstream-curvature-effect parameter that couples with the surface geometry inside the separated zone.

For the moderate curvature considered here, $\Delta \overline {p}_r$ and $\Delta \overline {p}_s$ are treated as small perturbation parameters. The scaled pressure-rise function can therefore be expanded about the flat-plate baseline as a first-order Taylor approximation. Retaining only the leading terms and neglecting higher-order and cross-contributions yields the additive form in (2.23). The coefficients $\alpha _r$ and $\alpha _s$ are then determined from the numerical database.

3.1. Experimental set-up

The experiments were conducted in the Mach 2.92 suction-type supersonic quiet wind tunnel at the National University of Defence Technology. The wind tunnel consists of a transition section, a rectifier section, a nozzle and experimental section, an expansion section, and a downstream vacuum tank. The experimental section of the wind tunnel is 400 mm in length, 200 mm in height, and 200 mm in width, and it is equipped with observation windows. The downstream vacuum tank has a volume of approximately 900 cubic metres, which allows the wind tunnel to operate stably for approximately 30 s. The wind tunnel draws air from the ambient atmosphere and accelerates it to Mach 2.92 through the nozzle. The freestream conditions in the experimental section are shown in table 1.

Table 1.

Inlet parameters of wind-tunnel test section.

Table 1. Long description

A table showing the freestream conditions in the experimental section of a wind tunnel. The table has one row and five columns. The column headers are M infinity, p infinity (Pa), T infinity (K), T 0 (K), and Re infinity (m^-1). The row contains the following values: 2.92, 3071, 110.1, 300, and 8.01 x 10^6 respectively.

In this study, the particle image velocimetry (PIV) method was employed to measure the velocity field, and wall pressure measurements were conducted using a pressure scanner through pressure taps. Titanium dioxide nanoparticles with diameter 42.5 nm were used as tracer particles. During the experiment, the tracer particles were injected into the wind tunnel using high-pressure gas. The laser has maximum pulse energy 520 mJ, and operates at frequency 2 Hz, with an inter-frame time set at 5 microseconds. Digital image pairs are acquired with an IMPERX charge-coupled device camera (4000 $\times$ 2672 pixel $^2$ ) equipped with a 105 mm f/2.8 D lens. The system is jointly controlled by a synchroniser with control accuracy 0.25 nanoseconds and a computer.

Wall static pressures were also acquired with a pressure-scanning valve. The measurement chain consists of a data-acquisition computer, a pressure scan module, and stainless steel tubes linking the module ports to the surface taps. Each channel can be sampled at up to 500 Hz with full-scale accuracy 0.05 %.

The test model consists of a $10^\circ$ wedge shock generator and a curved plate that produces a convex-wall boundary layer (figure 6). A 250 mm long flat segment upstream of the curved section is fitted with a 10 mm wide tripping strip 10 mm downstream of the plate leading edge to ensure a fully developed turbulent boundary layer. The curved panel has radius 2500 mm and extends 250 mm in the streamwise direction. The leading edge of the shock generator is located 185 mm downstream of the plate leading edge; its support height can be adjusted to impinge the shock at different streamwise positions.

A single row of pressure taps was drilled along the model centreline: 15 mm spacing on the flat section, and 3 mm spacing on the curved panel. Wall pressures were recorded with the scanning valve, and the resulting distributions were combined with PIV velocity fields to locate the separation and reattachment points, and to analyse the separated region.

Figure 6.

Schematic diagram of experimental model.

Figure 7.

Velocity profile of the upstream turbulent boundary layer obtained by PIV.

A tripping strip was placed at the leading edge of the flat plate to enforce boundary-layer transition. The flow parameters of the undisturbed boundary layer were obtained via PIV measurements, conducted at a streamwise position 250 mm downstream of the flat plate’s leading edge. The mean velocity distribution is illustrated in figure 7. To incorporate compressibility effects into the analysis, the van Driest effective velocity transformation (van Driest Reference van Driest1951) was adopted to extend incompressible boundary-layer theory to compressible flows, accounting for density variations through dimensional analysis adjustments. Combined with the Walz equation (Walz Reference Walz1969) and the ideal gas law under the assumption of constant pressure across the boundary layer, this transformation yields a closed-form solution for the modified velocity:

(3.1) \begin{align} u^* = \frac {u_e}{b} \arcsin \left ( \frac {2b^2(u/u_e) - a}{\sqrt {a^2 + 4b^2}} \right )\!, \end{align}

where $a = 0$ under the conditions of zero pressure gradient and adiabatic wall, and $ b$ is defined as

(3.2) \begin{align} b^2 = r \frac {\gamma - 1}{2} M_e^2 \frac {T_e}{T_w}, \end{align}

with $ r = Pr^{1/3} = 0.89$ representing the turbulent flow recovery factor, and $T_w$ taken as the freestream total temperature.

To cover the outer boundary-layer region, Coles’ wake law (Coles Reference Coles1956) was used for extension:

(3.3) \begin{align} U^+ = \frac {1}{0.41} \ln \left ( \frac {y u_\tau }{\nu } \right ) + 5.0 + \frac {2\varPi }{0.41} \sin ^2\left ( \frac {\pi }{2} \frac {y}{\delta } \right )\!. \end{align}

Here, $ \varPi$ denotes the wake strength parameter, $ \delta$ is the boundary-layer thickness, and the friction velocity $ u_\tau$ was calculated by least squares fitting of the transformed velocity profile to (3.3).

The transformed velocity profiles are shown in figure 7 in wall coordinates, together with the classical log law and Coles’ wake law. Good agreement between the experimental data and the theoretical curves is observed in both the logarithmic region and the outer wake region, demonstrating that the PIV measurements accurately capture the characteristic features of a turbulent boundary layer, and confirming that the incoming boundary layer is fully turbulent.

3.2. Numerical method

Although direct numerical simulations and large-eddy simulations have been applied to SWBLI, Reynolds-averaged Navier–Stokes (RANS) modelling remains the only practical option for generating large data sets and for engineering design, and it will continue to be the workhorse approach for the foreseeable future. Numerous studies have demonstrated that RANS can reproduce SWBLI with acceptable accuracy (Sekar et al. Reference Sekar, Karthick, Jegadheeswaran and Kannan2020; Jordan et al. Reference Jordan, Edwards and Stefanski2021; Zuo et al. Reference Zuo, Memmolo and Pirozzoli2021; Kolesnik et al. Reference Kolesnik, Smirnov and Smirnovsky2022; Zhou et al. Reference Zhou, Zhao, He, Zhao and Gao2023).

At present, most RANS studies of SWBLI still adopt the Spalart–Allmaras (S–A) or Menter shear stress transport (SST) $k{-}\omega$ models, which are adequate for many routine applications. However, both formulations exhibit clear deficiencies when applied to SWBLI. The complex shock system and massive separation inherent to SWBLI introduce large model-form uncertainties into conventional eddy-viscosity closures (Li et al. Reference Li, Zeng, Jiang, Li and Yan2022). For planar interactions, the S–A model reproduces the mean velocity and wall pressure distributions reasonably well (Zuo et al. Reference Zuo, Memmolo and Pirozzoli2021), but it is unable to capture Reynolds stress anisotropy – a quantity known to be essential for predicting separation at high Mach numbers (Raje & Sinha Reference Raje and Sinha2021); consequently, separation-length estimates obtained with S–A are unreliable. The Menter SST $k{-}\omega$ model, arguably the most popular two-equation closure, has been validated extensively for attached and mildly separated flows, yet for high-speed SWBLI, its structural coefficients do not reflect the true evolution of the flow, leading to significant scatter with respect to experimental data (Vyas et al. Reference Vyas, Yoder and Gaitonde2019; Boychev et al. Reference Boychev, Barakos and Steijl2020; Li et al. Reference Li, Zeng, Chen, Zhang and Yan2021; Raje & Sinha Reference Raje and Sinha2021; Li et al. Reference Li, Zeng, Jiang, Li and Yan2022; Zhang et al. Reference Zhang, Li, Zeng, Wang and Yan2022; Tang et al. Reference Tang, Li, Zeng, Li and Yan2023). Like S–A, SST $k{-}\omega$ also fails to reproduce the anisotropic Reynolds stresses generated across the shock (Raje & Sinha Reference Raje and Sinha2021).

The present work focuses on shock-induced separation over a curved wall, for which accurate prediction of both separation extent and wall pressure is essential. At high Mach numbers, Reynolds stress models (RSMs) have been shown to yield considerably better estimates of separation length than eddy-viscosity closures (Boychev et al. Reference Boychev, Barakos and Steijl2020; Zuo et al. Reference Zuo, Wei, Hu and Pirozzoli2022) – a prerequisite if the influence of surface curvature is to be quantified reliably. By solving individual transport equations for each Reynolds stress component, RSM formulations naturally capture anisotropy and provide a more faithful representation of turbulence/shock interactions (Huang et al. Reference Huang, Tan, Sun and Ling2016; Karl et al. Reference Karl, Hickey and Lacombe2017; Erb & Hosder Reference Erb and Hosder2020; Denggao et al. Reference Denggao, Fanzhi, Tianxin, Chen and Chao2023). Consequently, an RSM is adopted for the current simulations of a shock impinging on a curved boundary layer.

The RSM directly establishes differential equations for the turbulent fluctuating stresses in the RANS equations, namely the Reynolds stress transport equations:

(3.4) \begin{align} \frac {\partial \big(\overline {\rho }\overline {u'_{i}u'_{\kern-1pt j}}\big)}{\partial t} +\underbrace {\frac {\partial \big(\overline {\rho }\, \overline {u}_{k}\overline {u'_{i}u'_{\kern-1pt j}} \big)}{\partial x_{k}}}_{C_{\textit{ij}}\,\equiv\, \text{convection}} ={}& \underbrace {-\frac {\partial }{\partial x_{k}}\left ( \big(\overline {\rho }\overline {u'_{i}u'_{\kern-1pt j}u_k}\big) +\overline {p' u'_{i}}\delta _{kj} +\overline {p' u'_{\kern-1pt j}}\delta _{ik} \right )}_{D_{T,ij}\,\equiv\, \text{turbulent diffusion}} \nonumber \\[6pt]&{}+ \underbrace {\frac {\partial }{\partial x_k} \left [ \mu \frac {\partial \overline {u'_i u'_j}}{\partial x_k} \right ]}_{D_{L,ij}\,\equiv\, \text{molecular diffusion}} \underbrace {-\overline {\rho }\left ( \overline {u'_{i}u'_{k}}\frac {\partial \overline {u}_j}{\partial x_{k}} +\overline {u'_{\kern-1pt j}u'_{k}}\frac {\partial \overline {u}_i}{\partial x_{k}} \right )}_{P_{\textit{ij}}\,\equiv\, \text{stress production}} \nonumber \\[6pt]&{}+ \underbrace {\overline {p'\left ( \frac {\partial u'_{i}}{\partial x_{j}} +\frac {\partial u'_{\kern-1pt j}}{\partial x_{i}} \right )}}_{\varPhi _{\textit{ij}}\,\equiv\, \text{pressure strain}} -\underbrace {2\mu \overline { \frac {\partial u'_{i}}{\partial x_{k}} \frac {\partial u'_{\kern-1pt j}}{\partial x_{k}} }}_{\varepsilon _{\textit{ij}}\,\equiv\, \text{dissipation}} \nonumber \\[6pt]&{}- \underbrace {2\overline {\rho }\varOmega _k\left ( \overline {u'_j u'_m} e_{ikm} + \overline {u'_i u'_m} e_{jkm} \right )}_{F_{\textit{ij}}\,\equiv\, \text{production by system rotation}}\!. \end{align}

On the left-hand side of the equation, the first term represents the transient term, and the second term represents the convection term. On the right-hand side, the terms are sequentially the turbulent diffusion term, the molecular viscosity diffusion term, the stress production term, the pressure strain term, the viscous dissipation term, and the production by system rotation term, where $\varOmega _k$ is the system rotation vector. Here, we focus primarily on the modelling of the most important pressure strain term.

Given the focus of this paper on the boundary layer and separated flow over curved surfaces, the stress– $\omega$ model is selected to model the pressure strain term. This model is constructed based on the omega equation and the stress transport model of the LRR model (van Veen et al. Reference van Veen, Giepman, Schrijer and van Oudheusden2015; Vyas et al. Reference Vyas, Yoder and Gaitonde2019). It does not require modelling of the wall reflection term, and performs well in simulating flows with rotation. The expression for this model is

(3.5) \begin{align} \varPhi _{\textit{ij}} ={}&-C_1\rho \beta ^*_{\textit{RSM}}\omega \left ( \overline {u'_{i}u'_{\kern-1pt j}}-\frac {2}{3}k\delta _{\textit{ij}} \right ) -\frac {8+C_2}{11}\left (P_{\textit{ij}}-\frac {1}{3}P_{kk}\delta _{\textit{ij}}\right )\nonumber \\[6pt]&{}-\frac {8C_2-2}{11}\left (D_{\textit{ij}}-\frac {1}{3}P_{kk}\delta _{\textit{ij}}\right ) -k\frac {60C_2-4}{55}\left (S_{\textit{ij}}-\frac {1}{3}S_{kk}\delta _{\textit{ij}}\right )\!, \end{align}

where

(3.6) \begin{align} D_{\textit{ij}}=-\overline {\rho }\left (\overline {u'_{i}u'_{m}}\frac {\partial u_{m}}{\partial x_{j}}+\overline {u'_{\kern-1pt j}u'_{m}}\frac {\partial u_{m}}{\partial x_{i}} \right )\!. \end{align}

Here, $k$ is the turbulent kinetic energy, and $\omega$ is the specific dissipation rate. The mean strain rate tensor is defined as $S_{\textit{ij}}=(\partial \overline {u}_{i} /\partial x_{j} + \partial \overline {u}_{\kern-1pt j} /\partial x_{i})/2$ . The model constants are set to their standard values: $\beta ^*_{\textit{RSM}}=0.09$ , $C_1=1.8$ and $C_2=0.52$ . This model can accurately calculate the average flow characteristics and all components of the Reynolds stress in curved flow.

The numerical method employed in this study is the finite volume method. The inviscid fluxes are evaluated with the AUSM $^+$ scheme, which combines the high resolution of flux-difference splitting with the robustness of flux-vector splitting, and is therefore well suited for capturing shocks while preserving accuracy across contact discontinuities and boundary layers. The scheme separates the linear and nonlinear characteristic fields associated with the convective speeds, and treats the pressure and convective fluxes independently (Liou & Steffen Reference Liou and Steffen1993; Liou Reference Liou1996; Vitae et al. Reference Vitae, Fengxian and Vitae2003). Viscous fluxes are approximated by a second-order central difference. The MUSCL reconstruction with a min-mod limiter is used to achieve third-order accuracy in smooth regions while maintaining monotonicity near discontinuities. The ideal gas model is adopted, with specific heat ratio set at 1.4, and the viscosity coefficient is calculated using the Sutherland formula.

To systematically investigate the effects of inflow Mach number, shock strength, boundary-layer thickness and shock impingement location on the interaction characteristics, a series of numerical simulations were performed. Three inflow Mach numbers ( $M_\infty = 2.5$ , $3.0$ and $3.5$ ) and two shock-generator wedge angles ( $\theta _w = 12^\circ$ and $15^\circ$ , corresponding to flow deflection angles approximately $12^\circ$ and $15^\circ$ ) were considered. Different incoming boundary-layer thicknesses were obtained by varying the length of the flat plate upstream of the curved section ( $L = 250$ , $300$ and $350$ mm). The complete set of inflow conditions is summarised in table 2. The corresponding boundary-layer parameters at the reference locations (taken at the interaction onset $x_0$ ) are listed in table 3, including the incompressible and compressible definitions for displacement thickness $\delta ^*$ , momentum thickness $\theta$ , shape factor $H$ , and momentum thickness Reynolds number $Re_\theta$ . The incident shock impingement locations for all cases are provided in table 4.

Table 2.

Inflow conditions for numerical simulations. Here, $M_\infty$ is the freestream Mach number, $u_\infty$ is the freestream velocity, $\rho _\infty$ is the freestream density, $p_\infty$ is the freestream static pressure, $T_\infty$ is the freestream static temperature, $P_0$ is the total pressure, and $T_0$ is the total temperature. All cases share the same total pressure and total temperature, corresponding to a constant stagnation condition.

Table 2. Long description

A table summarizing inflow conditions for numerical simulations. The table has seven columns and three rows. The columns are labeled as follows: M infinity, u infinity (m s^-1), rho infinity (kg m^-3), p infinity (Pa), T infinity (K), P0 (kPa), and T0 (K). The rows present the following data: Row 1: M infinity, 2.5; u infinity (m s^-1), 578.48; rho infinity (kg m^-3), 0.1549; p infinity (Pa), 5930; T infinity (K), 133.33; P0 (kPa), 101; T0 (K), 300. Row 2: M infinity, 3.0; u infinity (m s^-1), 622.45; rho infinity (kg m^-3), 0.0897; p infinity (Pa), 2758; T infinity (K), 107.14; P0 (kPa), 101; T0 (K), 300. Row 3: M infinity, 3.5; u infinity (m s^-1), 654.04; rho infinity (kg m^-3), 0.0532; p infinity (Pa), 1328; T infinity (K), 86.96; P0 (kPa), 101; T0 (K), 300.

Table 3.

Turbulent boundary-layer parameters at the reference location (interaction onset $x_0$ ) for numerical simulations. Here, $u_e$ is the boundary-layer edge velocity, $\delta$ is the boundary-layer thickness (defined as $y$ where $u = 0.99u_e$ ), $\delta ^*$ is the displacement thickness, $\delta _{\theta }$ is the momentum thickness, $H = \delta ^*/\delta _{\theta }$ is the shape factor, and $Re_\theta = \rho _e u_e \delta _{\theta } / \mu _e$ is the momentum thickness Reynolds number. Two definitions are provided: the superscript ‘inc.’ denotes values computed using the incompressible formulation, while ‘com.’ denotes values obtained from the compressible formulation, which accounts for density variations across the boundary layer.

Table 3. Long description

A table comparing turbulent boundary-layer parameters for numerical simulations. The table has 12 rows and 7 columns. The columns are labeled as follows: M infinity, ue (m/s), delta (mm), delta* (mm), theta (mm), H, and Re_theta. The rows present values for different conditions. Row 1: M infinity, 2.5; ue (m/s), 578; delta (mm), 3.49; delta* (mm), 0.67 inc., 1.37 com.; theta (mm), 0.46 inc., 0.31 com.; H, 1.47 inc., 4.40 com.; Re_theta, 4463 inc., 3040 com. Row 2: M infinity, 2.5; ue (m/s), 4.27; delta (mm), 0.76 inc., 1.56 com.; theta (mm), 0.53 inc., 0.36 com.; H, 1.45 inc., 4.34 com.; Re_theta, 516 inc., 3530 com. Row 3: M infinity, 2.5; ue (m/s), 4.72; delta (mm), 0.86 inc., 1.78 com.; theta (mm), 0.60 inc., 0.41 com.; H, 1.43 inc., 4.34 com.; Re_theta, 584 inc., 4011 com. Row 4: M infinity, 3.0; ue (m/s), 622; delta (mm), 3.86; delta* (mm), 0.76 inc., 1.71 com.; theta (mm), 0.49 inc., 0.29 com.; H, 1.54 inc., 5.79 com.; Re_theta, 3750 inc., 2235 com. Row 5: M infinity, 3.0; ue (m/s), 4.49; delta (mm), 0.86 inc., 1.96 com.; theta (mm), 0.57 inc., 0.34 com.; H, 1.52 inc., 5.74 com.; Re_theta, 4329 inc., 2593 com. Row 6: M infinity, 3.0; ue (m/s), 4.97; delta (mm), 0.96 inc., 2.20 com.; theta (mm), 0.64 inc., 0.39 com.; H, 1.50 inc., 5.68 com.; Re_theta, 4893 inc., 2947 com. Row 7: M infinity, 3.5; ue (m/s), 654; delta (mm), 4.06; delta* (mm), 0.89 inc., 2.21 com.; theta (mm), 0.55 inc., 0.28 com.; H, 1.63 inc., 7.49 com.; Re_theta, 3223 inc., 1670 com. Row 8: M infinity, 3.5; ue (m/s), 4.72; delta (mm), 1.01 inc., 2.43 com.; theta (mm), 0.63 inc., 0.33 com.; H, 1.60 inc., 7.42 com.; Re_theta, 3708 inc., 1931 com. Row 9: M infinity, 3.5; ue (m/s), 5.49; delta (mm), 1.12 inc., 2.72 com.; theta (mm), 0.71 inc., 0.37 com.; H, 1.58 inc., 7.34 com.; Re_theta, 4180 inc., 2190 com.

Table 4.

Incident shock locations $x_{sc}$ (in mm) for different inflow conditions for numerical simulations. The values are measured along the wall from the flat/curved junction ( $x=0$ ) to the point where the inviscid shock impinges on the wall for each Mach number, shock-generator wedge angle $\theta _w$ and shock angle $\beta$ .

Table 4. Long description

A table comparing incident shock locations for different inflow conditions in numerical simulations. The table has 12 rows and 7 columns. The columns are labeled M infinity, theta w, beta, and x sc (mm). The rows present data for different combinations of Mach numbers, wedge angles, and shock angles. Each row lists specific values for these parameters. The values in the x sc (mm) column represent the shock impingement locations measured along the wall from the flat/curved junction to the point where the inviscid shock impinges on the wall. The table includes data for Mach numbers 2.5, 3.0, and 3.5, wedge angles 12 and 15 degrees, and corresponding shock angles. The values are presented in millimeters.

A grid independence study was conducted to prevent significant errors arising from the grid resolution and to avoid the computational waste or potential inaccuracies associated with an excessively dense mesh. The parameters of the different grids used for comparison are listed in table 5. Refinement was primarily applied to the near-wall region and the vicinity of the shock impingement area.

Table 5.

Parameters for grid independence verification.

Table 5. Long description

A table comparing mesh parameters for grid independence verification. The table has four columns and three rows. The columns are labeled Mesh, N y, Delta y plus min, N x, Delta x plus min, and N total. The rows are labeled Grid-A, Grid-B, and Grid-C. Row 1: Mesh, Grid-A; N y, 150; Delta y plus min, 0.89; N x, 660; Delta x plus min, 40.0; N total, 9.9 times 10 to the power of 4. Row 2: Mesh, Grid-B; N y, 200; Delta y plus min, 0.89; N x, 810; Delta x plus min, 30.2; N total, 1.62 times 10 to the power of 5. Row 3: Mesh, Grid-C; N y, 240; Delta y plus min, 0.45; N x, 840; Delta x plus min, 20.8; N total, 2.02 times 10 to the power of 5.

The computational model and grid are illustrated in figure 8. A wall-aligned natural coordinate system is employed, with the $x$ -axis tangential to the wall, the $y$ -axis normal to the wall, and the origin located at the junction between the flat and curved sections. The computational domain has height 80 mm. The shock generator has wedge length 60 mm. Two convex radii, $R = 250$ and $500$ mm, are considered, each with streamwise extent 120 mm along the curved wall, followed by a 50 mm straight segment.

The numerical results for the wall pressure distribution of the SWBLI flow field, which were obtained from the three grid resolutions, are compared in figure 9(a). The computed pressure distributions are generally similar across the different grids. However, the coarser Grid-A shows some deviation compared to the finer-resolution cases, while the results from the finer grids, Grid-B and Grid-C, exhibit negligible differences, indicating that the simulation results are no longer sensitive to further grid refinement. Therefore, the Grid-B configuration was selected for all subsequent numerical simulations in this study.

Figure 8.

Computational domain mesh and boundary conditions.

Figure 9.

(a) Comparison of wall pressure distributions between different grid resolution calculations of SWBLI. (b) The van Driest transformed mean velocity profile of turbulent boundary layer at the representative position as well as its comparison with that of Schlatter & Örlü (Reference Schlatter and Örlü2010) and Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000).

Figure 9(b) shows the van Driest transformed mean velocity profile of the undisturbed turbulent boundary layer at $x = 0$ . Here, $U^+$ is defined as

(3.7) \begin{align} U^+ = \int _0^{u/u_\tau } \sqrt {\frac {\rho }{\rho _w}} \, \mathrm{d}\! \left (\frac {u}{u_\tau }\right )\!. \end{align}

It can be seen that the transformed velocity profile agrees well with the law of the wall and the results of Schlatter & Örlü (Reference Schlatter and Örlü2010) and Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000). This agreement confirms that the present RANS approach accurately captures the turbulent boundary-layer characteristics upstream of the interaction.

The computational results obtained using different turbulence models are compared against the experimental data of Daub et al. (Reference Daub, Willems and Gülhan2015) in figure 10(a). The case selected for comparison has freestream Mach number 3 and wedge angle $17.5^\circ$ , with a geometrical set-up similar to that of the present study, and serves as a validation case for separated flows induced by relatively strong incident shocks. It is evident that the results from the stress– $\omega$ RSM agree most closely with the experimental measurements. The SST $k{-}\omega$ model underpredicts the plateau pressure within the separation region, while the S–A model shows a significant error in predicting the extent of the separation region, which is critically important for the present investigation.

Figure 10.

(a) Comparison of wall pressure distributions between calculation with different turbulence models and experimental data of Daub et al. (Reference Daub, Willems and Gülhan2015). (b) Comparison of wall pressure distributions between calculation with the stress– $\omega$ RSM using different shock-generator wedge lengths and present experimental measurement. (c) Comparison of velocity field between numerical calculation and experimental results.

In addition to the separated flow induced by a stronger shock shown in figure 10(a), an SWBLI generated by a weaker shock on the curved surface was also obtained using the present experimental set-up, and compared with numerical results. The experimental conditions are as described above. To assess the influence of the expansion waves emanating from the trailing edge of the wedge on the separated flow, an additional case with a different wedge length was included for comparison, as shown in figure 10(b), aiming to evaluate the effect of downstream expansion on the pressure distribution within the separation region. It can be observed that the peak wall pressure exhibits some deviation due to the expansion waves, while the pressure distribution inside the separation region remains largely unchanged, and the conditions near the reattachment point are very similar. The difference in separation length between the two wedge lengths is only 6 %, while the deviation between the case with wedge length 60 mm and the experimental result is also merely 8.7 % in figure 10(c). The numerical results show good agreement with the experimental data, particularly in terms of the pressure plateau.

In summary, the numerical methodology employed in this study is well suited for simulating SWBLI flow fields. It accurately captures both the wall pressure distribution and the velocity field structure of the separated flow, thereby satisfying the requirements for the present research.