2. Problem description and baseflow property

This investigation specifically examines the first Mack mode and cross-flow instabilities over a spanwise-infinite swept wing at zero angle of attack, as illustrated in figure 1. Here $x$ , $y$ and $z$ represent the Cartesian coordinates, and $s$ , $\eta$ and $z$ denote the local body-fitted coordinates, $\varLambda$ is the sweep angle, $U_\infty$ is the free stream velocity. Figure 1(b) illustrates the top view of the wing. Here $k_\infty$ is the free stream wavevector in the $x{-}z$ plane. It should be noted that the wave angle $\theta _\infty$ is defined between the wavevector $k_\infty$ and the free stream velocity $U_\infty$ to facilitate further calculations. The blue line represents an inviscid streamline, with the streamline-aligned velocity $u_s$ and the cross-flow velocity $u_c$ marked with yellow arrows. As depicted in figure 1(c), the geometry of the biconvex airfoil is an arc, designed to sustain quasiconstant pressure gradients, with a chord length of $200$ and a thickness of $1$ , where the length scales are non-dimensionalized by a reference length $L^*=0.001\, {\textrm{m}}$ . The $1:200$ thickness-to-chord ratio ensures moderate pressure gradient ranges, as the first Mack modes are fully damped under large favourable pressure gradients. This small thickness also results in relatively small discrepancies between $x$ and $s$ . The baseflow for a wing with a thickness of 0.5 is also simulated using identical numerical methods for LST analysis. Figure 1(d) illustrates the grid and boundary conditions for DNS. The DNS framework achieves resolution fidelity via very dense grids $1281 \times 501 \times 21$ ( $x \times y \times z$ ). The grid can be divided into three parts: the first part ranging from $x=-4$ to $40$ (red region marked in figure 1d) includes $881$ $x$ direction grid points to promise a very high grid resolution near the leading edge ensuring that the interactions between the boundary layer and free stream waves are captured exactly. This grid resolution promises at least 101 grid points for one wavelength in $x$ direction (for the vorticity wave case at $\varLambda =45^\circ$ ) and 21 grid points for one wavelength in $z$ direction. The second part, ranging from $x=40$ to $200$ , employs a medium grid resolution ( $381 \times 501 \times 21$ ) to establish a proper background baseflow field. The third part is the sponge region ranging from $x=200$ to $300$ (orange region marked in figure 1d), only employing a very coarse grid ( $21 \times 501 \times 21$ ). Grid independence is validated in the Appendix. The boundary conditions utilized in this study are shown in figure 1(d). At the inlet and upper surface, supersonic far-field boundary conditions are used for baseflow calculation, and free stream waves are superposed for receptivity calculation. Periodic conditions and supersonic outlet boundary conditions (with a coarse sponge layer) are used for the two lateral boundaries and the outlet of the domain, respectively. The lower surface before and after the wing surface ( $x \lt 0$ and ${\gt } 200$ ) are symmetry conditions. The wing surface is set to be a non-slip adiabatic wall.

The free stream conditions align with the experiment set-up of Stetson (Reference Stetson1983), which are summarized in table 1. The sweep angle $\varLambda$ ranges from $0^\circ$ to $45^\circ$ , enabling a systematic analysis of receptivity mechanism evolution across the first Mack mode and cross-flow instabilities.

Table 1.

Free stream conditions.

The governing equations for the DNS are the 3-D compressible Navier–Stokes equations, which can be written in the conservative form as

(2.1) \begin{align} \frac {\partial }{\partial t} \boldsymbol{Q}_i + \frac {\partial }{\partial x_j} \left ( \boldsymbol{F}_{ij} - \boldsymbol{F}_{{v} \boldsymbol{\cdot }ij} \right ) = 0 \end{align}

where $t$ and $x_j$ are time and the Cartesian coordinates, $\boldsymbol{Q}$ is the vector of conservative variables, $\boldsymbol{F}$ and $\boldsymbol{F}_v$ are inviscid and viscous flux vectors. The governing equations are closed by the equation of state of an ideal gas. The Prandtl number $Pr=0.72$ and the ratio of specific heats $\gamma =1.4$ . Sutherland’s law is used to obtain the dynamic viscosity $\mu$ (Hao et al. Reference Hao, Fan, Cao and Wen2022; Guo, Hao & Wen Reference Guo, Hao and Wen2025; Huang et al. Reference Huang, Cao, Hao, Guo and Wen2025). Spatial discretization applies the fifth-order upwind scheme for inviscid fluxes and the second-order central differencing scheme for viscous terms. The lower–upper symmetric Gauss–Seidel scheme is employed to resolve the steady flow field, and a third-order total variation diminishing Runge–Kutta scheme is used for unsteady temporal integration. This DNS code has been used to study the instabilities in hypersonic boundary layers in our previous research (Zhao et al. Reference Zhao, Wen, Tian, Long and Yuan2018, Reference Zhao, Wen, Long, Tian, Zhou and Wu2019; Liu et al. Reference Liu, Dong, Long, Zhao and Wen2023), and more detailed validations can be found in Lu et al. (Reference Lu, Uy, Zhao and Wen2025). The calculation process is stated as follows: the maximum residual reaches a small value of the order of $10^{-18}$ for the steady baseflow; then the unsteady free stream disturbances are superposed at the inlet and upper surface for simulating the leading-edge receptivity process. The detailed formations of the unsteady simulation and free stream waves are summarized in § 4.

In this paper, dimensional flow variables are non-dimensionalized by the steady-state free stream conditions. Specifically, length scales are non-dimensionalized by a reference length $L^*=0.001\,{\textrm{m}}$ , time scales by $L^*/U^*_\infty$ . The density, velocity, temperature and pressure are made dimensionless by the free stream parameters $\rho ^*_\infty$ , $U^*_\infty$ , $T^*_\infty$ and $\rho ^*_\infty U^{*2}_\infty$ . In the current study, the asterisk superscript ‘*’ typically signifies dimensional variables, whereas symbols without subscript ‘*’ represent non-dimensional variables.

Figure 2.

(a) The Mach number contour and (b) the pressure contour over the front airfoil ( $0\leqslant x \leqslant 40$ ) for the $\varLambda =0^\circ$ case.

Figure 3.

(a) Pressure and pressure gradients at the solid wall under different sweep angles; (b) boundary layer thickness and boundary layer edge Mach number under different sweep angles; (c) T profiles at $x=40$ ; (d) $u_c$ profiles at $x=40$ in swept cases.

Mach number distributions and pressure contours over the front airfoil ( $0\leqslant x \leqslant 40$ ) for the $\varLambda =0^\circ$ case are displayed in figure 2. A weak shock forms at the leading edge of the airfoil, followed by progressive pressure reduction through surface-generated expansion waves. Away from near-wall regions, Mach numbers exceed $5$ , maintaining hypersonic boundary layer conditions. Both first Mack mode and cross-flow instabilities are examined within the hypersonic regime. Figure 3(a) details wall pressure $p_w$ and streamwise pressure gradient $m$ distributions, where pressure gradient $m$ is calculated by

(2.2) \begin{align} m = \frac {1}{p_w} \frac {{\textrm{d}}p_w}{{\textrm{d}}s}. \end{align}

Based on this definition, negative $m$ values denote favourable pressure gradients. As shown in figure 3(a), the pressure exhibits a localized shift at the leading edge due to the shock wave, followed by a steady downstream decrease attributed to constant curvatures of the thin-wing configuration, with $m$ achieving near-constant values. Figure 3(b) depicts boundary layer thickness identified by the $0.99U_{\infty }$ criteria and boundary layer edge Mach number $Ma_e$ . The boundary layer thickness slightly increases as the sweep angle increases. Figure 3(c) depicts temperature profiles at $x=40$ . The cross-flow velocity $u_c$ , defined as the velocity component perpendicular to the inviscid streamline, constitutes a fundamental component of cross-flow instabilities (Lekoudis Reference Lekoudis1980; Su, Li & Han Reference Su, Li and Han2024). Figure 3(d) depicts cross-flow velocity profiles across all swept cases. Elevated sweep angles induce higher cross-flow velocities, suggesting that the cross-flow instability is more likely to dominate the transition process with increasing sweep angles. The maximum cross-flow velocity is 1.5 % of the free stream velocity. This low magnitude is attributed to the deliberately small thickness and weak chordwise pressure gradient. These cross-flow velocities fall below the threshold criterion established by Reed & Saric (Reference Reed and Saric1989). As their criterion was derived for low-speed flows, it is not applicable to the current configuration. The subsequent LST results will further confirm the existence of cross-flow instability.

3. Linear stability theory

Spatial LST is employed to capture the unsteady modes in this hypersonic flow. The vector $\boldsymbol{Q}$ from (2.1) is decomposed into a one-dimensional steady solution and an unsteady perturbation under the parallel flow assumption. After linearization, the perturbation term in normal-model form is written as follows:

(3.1) \begin{align} \boldsymbol{Q}^{\prime}(s, \eta , z, t) = \widehat {\boldsymbol{Q}}(\eta )\, \exp (i (\alpha _r+\alpha _i) s + i \beta z - i \omega t). \end{align}

Here $\alpha _r$ denotes the streamwise wavenumber, $\alpha _i$ denotes the spatial growth rate, $\beta$ is the spanwise wavenumber, $\omega$ is the circular frequency.

For the current research, the Reynolds number is large, and the boundary layer thickness is much smaller than the radius of curvature. The LST incorporating local curvature effects $\kappa$ is utilized for spatial analysis. The Lamé coefficients are $h_1=1+\kappa \eta$ and $h_2=h_3=1$ . For this configuration, omitting curvature terms introduces less than 2 % error in growth rates. Although the correction only yields a small difference on the LST result, curvature terms are retained for theoretical rigour and precise results. An eigenvalue problem is formulated by substituting (3.1) into the linearized compressible Navier–Stokes equations with $\boldsymbol{Q}$ ,

(3.2) \begin{align} \bar {A}\widehat {\boldsymbol{Q}} = (\alpha _r + i \alpha _i) \widehat {\boldsymbol{Q}}, \end{align}

where $\bar {A}$ represents the Jacobian matrix composed of the inviscid and viscous fluxes and is expressed in terms of base flow quantities. The detailed formulations can be found in prior studies (Ren & Fu Reference Ren and Fu2013; Song, Zhao & Huang Reference Song, Zhao and Huang2020). The LST code in this study has been fully validated in our previous investigations (Uy et al. Reference Uy, Hao and Wen2023a, Reference Uy, Hao and Wen2023b; Lu et al. Reference Lu, Uy, Zhao and Wen2025).

The spatial wavenumbers $\alpha _r$ and $\beta$ , and the spatial growth rate $\alpha _i$ are normalized by $L^*$ , while the circular frequency $\omega$ is normalized by $U^*_{\infty }/L^*$ . The transformation between dimensionless circular frequency ( $\omega$ ) and dimensional frequency ( $F^*$ ), dimensionless spanwise wavenumber ( $\beta$ ) and dimensional spanwise wavelength ( $\lambda ^*_z$ ) are

(3.3) \begin{align} \omega =\frac {2\pi \times F^* \times L^*}{|U^*_\infty |} \quad \beta =\frac {2\pi \times L^*}{\lambda ^*_z}. \end{align}

The LST will also be performed in the streamline-aligned coordinate, where velocity components are transformed into the localized inviscid streamline coordinates ( $u_s$ and $u_c$ in figure 1b). Subscript ‘ $str$ ’ represents quantities in streamline-aligned coordinates. Here $\beta _{str}$ is calculated by projecting the total spatial wavenumber to the cross-flow direction. Note that the use of streamline-aligned coordinates is valid only within the context of local stability analysis for illustrating the symmetric and asymmetric characteristics of the first Mack mode and cross-flow mode.

Figure 4.

Growth rate contours at $s=40$ for (a) the $\varLambda =0^\circ$ case in Cartesian coordinates and (b) the $\varLambda =45^\circ$ case in Cartesian coordinates, (c) the $\varLambda =45^\circ$ case in streamline-aligned coordinates and (d) the $\varLambda =45^\circ$ case in streamline-aligned coordinates with $u_c$ deliberately set to 0. Black solid lines, contour lines at $-\alpha _i$ = 0.025; black dashed lines, $\beta$ = 0 positions.

Figure 4 illustrates the spatial growth rate contours at $s=40$ for the $\varLambda =0^\circ$ case in Cartesian coordinates, the $\varLambda =45^\circ$ case in Cartesian coordinates, the $\varLambda =45^\circ$ case in streamline-aligned coordinates and the $\varLambda =45^\circ$ case in streamline-aligned coordinates with $u_c$ deliberately set to 0. Contour lines at $-\alpha _i=0.025$ are plotted to facilitate identifying different instability regions. Positions at $\beta =0$ are marked as dashed lines to enable observing the symmetry of instabilities.

The instability mechanism is first discussed within the Cartesian coordinate system, where velocity components $u$ and $w$ are employed. Since the DNS is applied in Cartesian coordinates, this LST approach enables direct comparisons in subsequent analyses. For $\varLambda =0^\circ$ , two instability regions, which are symmetric to $\beta =0$ , span extensive ranges of $\beta$ values within the lower-frequency domain. These instability regions correspond to the first Mack mode, as expected for 2-D configurations. At $\varLambda =45^\circ$ in Cartesian coordinates, three distinct instability regions are identified: Region 1 spans a wide range of $\beta$ and connects to zero-frequency branches, typifying the emergence of cross-flow instability in the traditional terminology; Region 2, a modified mode, is confined to a narrow $\beta$ region. This instability originates from the coalescence of the negative spanwise wavenumber of the first mode under the asymmetric effect of the 3-D baseflow, consistent with observations in Liu (Reference Liu2022) and Qiu et al. (Reference Qiu, Shi, Zhu and Lee2024). The asymmetry between Regions 1 and 2 also highlights the emergence of cross-flow instability. Region 3, typifying the oblique second Mack mode, occupies the high-frequency domain.

The LST results in the streamline-aligned coordinate are then analysed to clarify the mechanisms of each instability region. For the $\varLambda =45^\circ$ case in the streamline-aligned coordinate, Regions 1 and 2 remain asymmetric. Region 3 still characterizes the second Mack mode, with its maximum growth rate shifted away from $\beta =0$ . In figure 4(d), the cross-flow velocity is artificially set to 0, thereby illustrating the instability originating from the streamwise velocity. Compared with the $\varLambda =0^\circ$ case, the first mode becomes less significant and the second Mack mode shifts to lower-frequency regions. Cross-flow components universally destabilize the positive- $\beta$ branch, while stabilizing the negative- $\beta$ branch, confirming that cross-flow components act as a modulating mechanism rather than constituting a distinct instability mode.

Figure 5.

(a) Pressure gradients and (b) spatial growth rates at $x=40$ in cases of different thicknesses. Solid line with filled symbols are in Cartesian coordinates; dashed line with hollow symbols are in streamline-aligned coordinates.

As discussed before, favourable pressure gradients suppress first Mack modes (Zurigat et al. Reference Zurigat, Nayfeh and Masad1992; Masad & Zurigat Reference Masad and Zurigat1994), while amplifying cross-flow modes (Saric & Reed Reference Saric and Reed2003). A comparative analysis employs a decreased airfoil thickness of $0.5$ . Pressure gradients and spatial growth rates from LST at $x=40$ for both thickness configurations and in both coordinates are summarized in figure 5. As depicted in figure 5(a), thinner profiles substantially decrease favourable pressure gradients across all sweep angles. Results in both coordinates reveal the same trends: at $\varLambda =0^\circ$ and $15^\circ$ , the spatial growth rate diminishes with strengthened pressure gradients. For $\varLambda =30^\circ$ and $45^\circ$ , the instability is amplified under an enhanced favourable pressure gradient, confirming the predominance of the cross-flow instability. Based on the instability’s response to pressure gradients, the dominant mechanism is the first Mack mode when $\varLambda =0^\circ$ and $15^\circ$ , whereas the cross-flow instability for the $\varLambda =30^\circ$ and $45^\circ$ cases.

Although disturbances at distinct $\varLambda$ exhibit differential responses to pressure gradients, both oblique first modes and cross-flow modes belong to S-modes based on LST. Figure 6 presents eigenspectra at $s$ = 2, 4, 6, 8, 10, 15, 20, 25, 30, 35 and 40 for both $\varLambda =0^\circ$ and $\varLambda =45^\circ$ cases. Solid black trajectories track the evolution of the S-modes from stable to unstable states with increasing $s$ , while green circles highlight persistently neutral F-modes. This consistent behaviour in both cases indicates a fundamental connection between these instabilities. Table 2 documents neutral point locations where the first Mack or cross-flow modes’ growth rates transition to positive values. The neutral point shifts upstream with increasing sweep angles.

Table 2.

Locations of neutral points for the selected mode at different sweep angles.

Figure 6.

The evolution of F- and S-modes in (a) the $\varLambda =0^\circ$ case, (b) the $\varLambda =45^\circ$ case and (c) caption for both cases.

4. Receptivity to free stream waves

This section investigates the boundary layer receptivity mechanisms to free stream acoustic and vorticity perturbations at a circular frequency of $\omega = 0.485\rm$ and a spanwise wavenumber of $\beta = 1.571$ . As established in the prior analysis, S-modes with (without) cross-flow components serve as the dominant instability under these parametric conditions.

For unsteady calculations, small perturbations are superposed on the far-field boundary condition to simulate the receptivity process. The wave field for the receptivity simulation is expressed as

(4.1) \begin{align} \boldsymbol{Q}^{\prime}(x, y, z, t) = \boldsymbol{Q}(x, y, z, t) - {\boldsymbol{\bar{Q}}}(x, y, z) \end{align}

where $\boldsymbol{Q}^{\prime}(x,y,z,t)$ is the perturbation of a flow variable, $\boldsymbol{Q}(x,y,z,t)$ represents the instantaneous value from unsteady DNS and ${\boldsymbol{\bar{Q}}}(x,y,z)$ corresponds to the time-averaged values from steady DNS. The computational framework assumes spanwise-periodic conditions for swept-wing geometries, with free stream disturbance wave propagation occurring exclusively in the $x{-}z$ plane. For small-amplitude free stream perturbations, the perturbations of an arbitrary flow variable can be described in the following form:

(4.2) \begin{align} \begin{bmatrix} u^{\prime} \\ w^{\prime} \\ p^{\prime} \\ \rho ^{\prime} \end{bmatrix}_{\infty } = \begin{bmatrix} \left | u^{\prime} \right | \\ \left | w^{\prime} \right | \\ \left | p^{\prime} \right | \\ \left | \rho ^{\prime} \right | \end{bmatrix}_{\infty } \,\exp \left ( i \left [ \alpha _{r,\infty } x + \beta z - \omega t \right ] \right ), \end{align}

where $|u^{\prime}|$ , $|w^{\prime}|$ , $|p^{\prime}|$ and $|\rho ^{\prime}|$ represent dimensionless amplitudes of the free stream perturbations in Cartesian coordinates. The dispersion relations and perturbation magnitudes for the slow acoustic waves and vorticity waves are represented as follows (align with the definition in Ma & Zhong (Reference Ma and Zhong2005), the usage of different axis directions causes differences on the sign):

slow acoustic waves ( ${\textit{AC}}$ ), $\omega =k_\infty (-\frac {1}{M_\infty }+\cos \theta _\infty )$ ,

(4.3) \begin{equation} \begin{aligned} |u^{\prime}| &= -A \frac {M_\infty \alpha _{r,\infty }}{k_\infty }, \quad |w^{\prime}| = -A \frac {M_\infty \beta }{k_\infty },\\ |p^{\prime}| &= A, \quad \quad \quad \quad |\rho ^{\prime}| = A M_\infty ^2; \end{aligned} \end{equation}

vorticity waves ( ${\textit{VO}}$ ), $\omega =k_\infty \cos \theta _\infty$ ,

(4.4) \begin{equation} \begin{aligned} |u^{\prime}| &= A\frac {\beta }{k_\infty }, \quad |w^{\prime}| = -A\frac {\alpha _{r,\infty }}{k_\infty },\\ |p^{\prime}| &= 0, \quad \quad \quad |\rho ^{\prime}| = 0 .\end{aligned} \end{equation}

Hereinafter, the case nomenclature combining disturbance type and sweep angle, ${\textit{AC}}-0^\circ$ , as an example, corresponds to the slow acoustic wave case under $\varLambda =0^\circ$ . Here $A$ is the dimensionless magnitude of free stream waves, set to $10^{-6}$ in this study. The relationships between the free stream wavenumber vector $k_{\infty }$ , the incident wave angle $\theta _{\infty }$ (see figure 1), $\alpha _{r,\infty }$ and $\beta$ are

(4.5) \begin{equation} \begin{aligned} k_{\infty }=\sqrt {\alpha _{r,\infty }^2+\beta ^2} \quad \tan(\theta _{\infty }+\varLambda )=\beta /\alpha _{r,\infty }. \end{aligned} \end{equation}

The current research employs perturbations with identical frequencies ( $\omega$ ) and spanwise wavenumbers ( $\beta$ ) to enable systematic comparison, where $\omega$ and $\beta$ are first derived from LST in § 4. The undetermined quantities, i.e. $\alpha _{r,\infty }$ , $k_{\infty }$ and $\theta _{\infty }$ , are resolved by combining (4.5) with either (4.3) or (4.4). Table 3 catalogues computed $\alpha _{r,\infty }$ values across all configurations, where negative values indicate that trace velocities of the waves propagate opposite to the chordwise direction. It is important to reiterate that $\alpha _{r,\infty }$ and $\beta$ are defined in the chordwise and spanwise directions. When projecting $k_{\infty }$ onto the free stream velocity direction, vorticity waves always propagate with the free stream direction, not against it. After resolving these values, the free stream perturbations are introduced to the flow field by superposing the perturbations on the steady free stream variables at the far-field boundary, forming the free stream wave boundary condition. Note that linear growth phases occur upstream of $x=40$ in all cases. Unlike the second Mack mode, neither the first Mack nor cross-flow instabilities exhibit a synchronization point between F- and S-modes, and no new flow structures emerge after linear growth onsets. Receptivity calculations are hereby restricted to the front airfoil regions ( $x \leqslant 40$ , the first region of the grid described in § 2) to reduce calculation cost. As the perturbations are introduced at the boundaries, 3-D unsteady DNS simulations are performed with a small time step at $1.079 \times 10^{-4}$ , yielding 120 000 steps for one temporal period of the targeted perturbation. The sampling starts after $t=150$ per 6000 steps, yielding 21 samples within a temporal period. As the study focuses on the linear receptivity process, all the dynamics are periodic, making 21 samples enough for the Fourier transformation. The time series of fluctuating pressure at $z=0$ of the wall is processed by the Fourier transformation to obtain the phase angle $\phi ^{\prime}$ . The local wavenumber $\alpha _r$ of wall pressure disturbances is subsequently calculated by (Hao & Wen Reference Hao and Wen2021)

(4.6) \begin{equation} \alpha _r=L^* \frac {{\textrm{d}} \phi ^{\prime}}{{\textrm{d}}x} .\end{equation}

Based on the samples, the maximum wall pressure perturbation $p^{\prime}_{w,\textit{max}}$ is defined as the maximum absolute pressure fluctuation at the wall ( $\eta=0$ ) over one perturbation period and across the entire spanwise direction,

(4.7) \begin{equation} p_{w,\textit{max}}^{\prime }(x)=\textit{max}_{t}\left (\textit{max} _{z}\left (\left |p^{\prime}(x,\eta =0,z,t)\right |\right )\right ). \end{equation}

This quantity captures the most unstable disturbance at each streamwise station, thereby representing the evolution of unstable modes.

Table 3.

The spatial wavenumbers $\alpha _{r,\infty }$ of free stream disturbances.

After obtaining the perturbation field with unsteady DNS, MPT is applied on the perturbation fields for a physics-based interpretation of the receptivity process. By Helmholtz’s theorem, Doak’s MPT approach selects the momentum density $\boldsymbol{m}(\equiv \rho \boldsymbol{u})$ as the primary dependent vector field to be decomposed. The fluctuation momentum density $\boldsymbol{m}^{\prime }$ is decomposed into the vortical component $\boldsymbol{m_B^{\prime}}$ , acoustic component $\boldsymbol{m_A^{\prime}}$ and thermal component $\boldsymbol{m_T^{\prime}}$ as follows:

(4.8) \begin{equation} \boldsymbol{m}^\prime =\boldsymbol{m_B^\prime} +\boldsymbol{m_A^\prime} +\boldsymbol{m_T^\prime} . \end{equation}

The thermal component $\boldsymbol{m_T^{\prime }}$ is calculated by

(4.9) \begin{equation} \boldsymbol{m_T^{\prime}}=-\boldsymbol{\nabla} ^{2}\psi _{T}^{\prime }=\frac {(\gamma -1)\rho }{\gamma R}\frac {\partial S^{\prime }}{\partial t},\end{equation}

where $S^{\prime }$ is the entropy fluctuations. The acoustic component $\boldsymbol{m^{\prime }_A}$ is calculated by

(4.10) \begin{equation} \boldsymbol{m^{\prime }_A}=-\boldsymbol{\nabla} ^{2}\psi _{A}^{\prime },\quad\psi _{A}^{\prime }=\psi ^{\prime }-\psi _{T}^{\prime },\quad\boldsymbol{\nabla} ^{2}\psi ^{\prime }=\frac {\partial \rho ^{\prime }}{\partial t} .\end{equation}

The vortical component $\boldsymbol{m_B^{\prime}}$ is subsequently calculated by (4.8). All the equations are calculated by substituting the complex perturbation vectors obtained from the Fourier transformation on the perturbation field from unsteady DNS.

These MPT components exhibit distinct physical natures. The vortical component $\boldsymbol{m_B^{\prime}}$ is isobaric and isentropic. The acoustic component $\boldsymbol{m_A^{\prime}}$ is irrotational and isentropic. The thermal component $\boldsymbol{m_T^{\prime}}$ is irrotational and isobaric. Such orthogonal characteristics enable MPT to systematically dissect perturbation dynamics by isolating each physical component. Comprehensive derivations of the MPT decomposition can be found in our earlier article (Lu et al. Reference Lu, Uy, Zhao and Wen2025). The momentum density components are non-dimensionalized by the free stream momentum density $\rho ^*_\infty U^*_\infty$ .

4.1. Receptivity to slow acoustic waves

The spatial evolution of perturbation fields is systematically examined through density $\rho ^{\prime}$ and velocity $u^{\prime}$ distributions shown in figures 7 and 8. Figure 7 displays slices taken at $z=0$ , streamwise stations $x=1$ , $10$ , $20$ , $30$ , $40$ and the wall, while figure 8 details instantaneous snapshots of $u^{\prime}$ at $z=0$ and $x=30$ . Shock waves and boundary layer edges are marked by white solid lines and dashed lines in figure 8. Following Balakumar’s analytical framework (Balakumar Reference Balakumar2015), the whole perturbation field comprises four distinct regions: the free stream disturbance region; the shock layer; the area between the shock and boundary layer; the boundary layer.

Figure 7.

The density fluctuation contours of (a) ${\textit{AC}}-0^\circ$ , (b) $AC-15^\circ$ , (c) $AC-30^\circ$ and (d) $AC-45^\circ$ cases.

Figure 8.

The streamwise velocity fluctuation contours of (a,b) ${\textit{AC}}-0^\circ$ , (c,d) $AC-15^\circ$ , (e,f) $AC-30^\circ$ and (g,h) $AC-45^\circ$ cases at (a,c,e,g) $z=0$ and (b,d,f,h) $x=30$ .

The first region, located outside the shock, is characterized by uniformly propagating acoustic waves, as shown in figure 7. For slow acoustic waves, the magnitude of $\rho ^{\prime}$ is at the same level as that of the boundary layer disturbance, while $u^{\prime}$ is relatively small compared with the boundary layer values and consequently absent in figure 8. This originates from the magnitude relationship in (4.3). The second region encompasses the shock layer, where interactions between shock waves and slow acoustic waves result in localized amplifications of $\rho ^{\prime}$ . However, $u^{\prime}$ disturbances remain too weak to be visible in figure 8. The third region between the shock and the boundary layer comprises transmitted external acoustic fields and disturbances radiating from the boundary layer. These transmitted waves soon dissipate, whereas the radiating disturbances exhibit continuous amplifications as propagating downstream. Interestingly, figure 7(c) reveals exclusively positive $\rho ^{\prime}$ values across these three regions at $z=0$ . This is attributed to the small $\alpha _r$ value (as shown in table 3), yielding elongated streamwise wavelengths for free stream disturbances. Free stream disturbances still propagate uniformly in this case, which can be observed in the spanwise direction from the streamwise slices, for example, $x=30$ . The behaviours of acoustic waves and transmitted waves in these three regions align with 2-D receptivity simulations in Balakumar (Reference Balakumar2015) for acoustic waves, while diverging behaviours in the boundary layer arise due to distinct instability modes. Disturbances evolve and manifest as single-cell structures in the $\rho ^{\prime}$ and $u^{\prime}$ fields in the boundary layers. In all scenarios, the boundary-layer disturbances start from the leading edge and rapidly surpass free stream amplitudes. The disturbances within the boundary layer are critical to the transition process; therefore, they will be examined in detail through the analysis of wall pressure fluctuations.

Figure 9.

The wall pressure fluctuation amplitudes and streamwise wavenumbers for (a) ${\textit{AC}}-0^\circ$ , (b) $AC-15^\circ$ , (c) $AC-30^\circ$ and (d) $AC-45^\circ$ cases.

Figure 10.

The $u^{\prime}$ profiles for ${\textit{AC}}-0^\circ$ and $AC-45^\circ$ cases at neutral points.

The amplitudes and wavenumbers of wall pressure disturbances from DNS are presented in figure 9, compared with LST results. The maximum wall pressure disturbances $p^{\prime}_{w,max}$ from the DNS are plotted, along with N factors from LST for comparison. The N factors follow the definition in van Ingen (Reference van Ingen2008):

(4.11) \begin{equation} N=\int _{x_\textit{neutral}}^x-\alpha _i {\textrm{d}}x=\ln(A_x/A_0). \end{equation}

The N factor is calculated by integrating the growth rate from the neutral point $x_\textit{neutral}$ to streamwise location $x$ in LST. This equals the natural logarithm of the ratio of the disturbance amplitude $A_x$ to an initial amplitude $A_0$ . This relationship enables direct comparison between $p^{\prime}_{w,\textit{max}}$ and N factors: for case ${\textit{AC}}-0^\circ$ , the $p^{\prime}_{w,\textit{max}}$ axis spans $1 \times 10^{-8}$ to $2 \times 10^{-7}$ , corresponds to an N-factor range $\ln(2 \times 10^{-7}/1 \times 10^{-8}) \approx 2.996$ . The N factor is plotted on a linear scale preserving this range, with $x \gt 23$ aligned with $p^{\prime}_{w,\textit{max}}$ , confirming the onset of linear growth. This approach enables visual comparison between DNS and LST results. The N factor axis is then hidden for figure conciseness. The formation of the targeted S-modes goes through: Stage I, a leading-edge damping stage; Stage II, an enhanced exponential growth stage; and finally Stage III, the linear growth stage. The leading-edge damping and enhanced exponential growth stage are separated by the minimum value of $p^{\prime}_{w,\textit{max}}$ . Farther downstream, the emergence of a linear growth stage is defined as the location where the DNS and LST results are highly aligned. For oscillating $\alpha _r$ from DNS, it should be closely around LST values. Dark grey lines in figure 9 mark the onsets of enhanced exponential growth and linear growth.

The results indicate first Mack and cross-flow modes undergo a uniform receptivity pathway. At the leading edges, the interactions between acoustic waves, shock waves and boundary layers induce sudden amplitude enlargement followed by rapid damping, accompanied by abrupt $\alpha _r$ shift. After the local minimum of $p^{\prime}_{w, \textit{max}}$ , perturbation growth rates surpass LST predictions, represented by the larger slopes from DNS, along with the wavenumbers gradually converge towards S-mode values, forming an enhanced exponential growth stage. During this region, wavenumbers approach S-mode values sooner than the amplitudes, implying the perturbations possess some key features of the S-mode before the onset of linear growth. This is corroborated by $u^{\prime}$ -profiles at the neutral point (as depicted in figure 10), where cases ${\textit{AC}}-0^\circ$ and $AC-45^\circ$ exhibit strong agreement with LST eigenfunctions prior to linear growth phases. After enhanced exponential growth, both spatial growth rates and wavenumbers of perturbations align with LST solutions, indicating linear growths of the targeted S-modes occur. Two fundamental questions arise from these observations: Why does rapid damping occur at the leading edge despite the synchronization between S-modes and slow acoustic waves? How does the enhanced exponential growth happens when perturbation profiles already approximate S-mode eigenfunctions? The underlying mechanisms of this receptivity pathway are then discussed with MPT.

Figure 11.

The instantaneous snapshots of (a,b) acoustic components, (c,d) vortical components, (e,f) and thermal components in (a,c,e) ${\textit{AC}}-0^\circ$ and (b,d,f) $AC-45^\circ$ cases.

Figure 12.

The normalized amplitudes of (a,d) acoustic components, (b,e) vortical components, and (c,f) thermal components in ${\textit{AC}}-0^\circ$ (a,b,c) and $AC-45^\circ$ (d,e,f) cases.

Instantaneous snapshots of MPT component, defined as $\boldsymbol{m^{\prime}_{A, B or T}}=Re(m^{\prime}_{A, B or T, x})+Re(m^{\prime}_{A, B or T, y})+Re(m^{\prime}_{A, B or T, z})$ , in ${\textit{AC}}-0^\circ$ and $AC-45^\circ$ cases are illustrated in figure 11. Shock waves and boundary layer edges are marked by white solid lines and dashed lines, whereas the vertical black dashed line indicates the beginning of enhanced exponential growth and linear growth stages. Momentum potential theory components exhibit distinct behaviours in different regions: in the free stream disturbance region, the acoustic component $\boldsymbol{m_A^{\prime}}$ propagates uniformly, whereas $\boldsymbol{m_B^{\prime}}$ and $\boldsymbol{m_T^{\prime}}$ remain negligible in the free stream. At the shock layer, $\boldsymbol{m_A^{\prime}}$ experiences localized distortion, accompanied by weak $\boldsymbol{m_B^{\prime}}$ generation due to baroclinity, while $\boldsymbol{m_T^{\prime}}$ persists at negligible levels. The postshock transmitted waves are composed of acoustic and vortical waves.

Figure 12 depicts the amplitudes of MPT components at $s=1, 20, 40$ , aiming to capture the leading-edge damping, enhanced exponential growth and linear growth behaviours. Each amplitude is normalized by the maximum fluctuation momentum density $\boldsymbol{m}^{\prime }$ in boundary layers. During the linear growth stage (the $s=40$ profile), $\boldsymbol{m_B^{\prime}}$ is the dominant component, exceeding both $\boldsymbol{m_A^{\prime}}$ and $\boldsymbol{m_T^{\prime}}$ in magnitudes by orders, which reflects the vortical nature of the targeted modes. This vortical-dominant behaviour aligns with our prior findings for cross-flow instabilities (Lu et al. Reference Lu, Uy, Zhao and Wen2025).

For the disturbance evolution in boundary layer regions, the receptivity stages is explained by MPT component dynamics. At the leading-edge, significant $\boldsymbol{m_A^{\prime}}$ is generated through the synchronization mechanism (Xu et al. Reference Xu, Liu, Chen and Fu2018). Here $\boldsymbol{m}^{\prime }$ rapidly decays as this large acoustic component cannot persist in boundary layers. Concurrently, $\boldsymbol{m_B^{\prime}}$ makes up a smaller proportion than the linear stage, while $\boldsymbol{m_T^{\prime}}$ constitutes a large fraction. The high amplitude and fast dissipation at the leading edge observed in figure 9 correspond to the damping of acoustic perturbations. During the enhanced exponential growth stage, $\boldsymbol{m_A^{\prime}}$ dissipates to a small proportion, with $\boldsymbol{m_T^{\prime}}$ reducing to a lower level than the linear growth. In figures 11(c,d) and 12(b,e), $\boldsymbol{m_B^{\prime}}$ exhibits continuous growth in magnitudes from $s=1$ to $20$ . The predominance of $\boldsymbol{m_B^{\prime}}$ near the neutral point results in the good agreement between perturbation profiles and LST eigenfunctions in figure 10, since the vortical component is the primary contributor to the targeted S-modes. In this stage, the adjustment of $\boldsymbol{m_A^{\prime}}$ and $\boldsymbol{m_T^{\prime}}$ drive $p^{\prime}_{w,\textit{max}}$ and $\alpha _r$ towards convergence with LST results, whereas the high growth rate is credited to the growth of $\boldsymbol{m_B^{\prime}}$ . Farther downstream, all MPT components enter a steady growth phase, signalling the free stream acoustic waves successfully excite the targeted S-modes, thereby completing the receptivity process.

Both ${\textit{AC}}-0^\circ$ and $AC-45^\circ$ cases share the same receptivity mechanisms, with their divergence manifesting in amplitudes of MPT components. The underlying mechanisms of the receptivity pathway can be summarized as: $\boldsymbol{m_A^\prime}$ initially penetrates the boundary layer at high proportions, subsequently undergoing damping and recovery; $\boldsymbol{m_B^\prime}$ exhibits continuous growth throughout the receptivity process; $\boldsymbol{m_T^\prime}$ goes through a similar pathway with $\boldsymbol{m_A^\prime}$ but with lower initial proportions among the three components.

The receptivity coefficient refers to the efficiency of disturbance waves in generating instability waves, thus is a crucial factor in engineering. According to Balakumar’s definition (Balakumar & King Reference Balakumar and King2011), the receptivity coefficient to slow acoustic waves is defined at the neutral point, formed as

(4.12) \begin{align} C_{AC}=\left (u^{\prime }_{\textit{max} } / |p^{\prime }|_{AC} \right )_\textit{neutral} .\end{align}

As neutral points lie in enhanced exponential growth stages, where dynamics are governed by the behaviour of increasing $\boldsymbol{m_B^\prime}$ , this metric effectively quantifies the efficiency of acoustic waves in exciting vortical components, offering practical insight for engineering applications. The receptivity coefficients to slow acoustic waves for various sweep angles are summarized by the red line in figure 13. Here $C_{AC}$ demonstrates a consistent decrease as the sweep angle increases. The absence of abrupt discontinuities in $C_{AC}$ when the sweep angle increases further highlights that these two instabilities are of the same nature.

Figure 13.

The receptivity coefficients of all cases.

Figure 14.

The streamwise velocity fluctuation contours of (a) ${\textit{VO}}-0^\circ$ , (b) ${\textit{VO}}-15^\circ$ , (c) ${\textit{VO}}-30^\circ$ and (d) ${\textit{VO}}-45^\circ$ cases.

Figure 15.

The density fluctuation contours of (a,b) ${\textit{VO}}-0^\circ$ , (c,d) ${\textit{VO}}-15^\circ$ , (e,f) ${\textit{VO}}-30^\circ$ and (g,h) ${\textit{VO}}-45^\circ$ cases at (a,c,e,g) $z=0$ and (b,d,f,h) $x=30$ .

4.2. Receptivity to vorticity waves

Consistent with (4.4), vorticity waves are introduced into the flow field. Figure 14 illustrates the spatial distributions of $u^{\prime}$ across all cases with vorticity disturbances. White solid lines are plotted in the $z=0$ slice to mark the shock wave positions. Instantaneous snapshots of $\rho ^{\prime}$ at $z=0$ and $x=30$ are presented in figure 15. Shock waves and boundary layer edges are marked by white solid lines and dashed lines. Given the unique receptivity behaviour associated with vorticity waves, perturbation fields near leading edges are presented in zoomed-in views to facilitate observation. The flow field is also divided into four regions: the free stream disturbance region; the shock layer; the area between the shock and boundary layer; the boundary layer.

In the free stream disturbance regions, only velocity disturbances ( $u^{\prime}$ ) are observed, while $\rho ^{\prime}$ remain absent, consistent with the governing relation defined in (4.4). Shock layers generate density disturbances at a comparable order of magnitude to those in the boundary layer, while changes in $u^{\prime}$ are small. Impacts of weak leading-edge shock waves on vorticity disturbances are less significant than that on slow acoustic waves. The transmitted waves closely resemble vorticity waves and connect to boundary layers, which complicate the identification of unstable modes with $u^{\prime}$ in boundary layers. A practical method for identifying unstable modes is to observe $\rho ^{\prime}$ in figure 15. Disturbances with short spatial wavelengths are generated through the interaction between shock waves, vorticity waves and boundary layers at the leading edge. These disturbances dissipate quickly, forming ‘dead zones’, which are characterized as the relatively low magnitudes of perturbations. Downstream of these ‘dead zones’, the targeted S-modes are established. Within a considerable distance (until approximately $x=30$ ), $u^{\prime}$ amplitudes within boundary layers remain smaller than the free stream perturbation levels (see figure 14).

Figure 16.

The wall pressure fluctuation amplitudes and streamwise wavenumbers for (a) ${\textit{VO}}-0^\circ$ , (b) ${\textit{VO}}-15^\circ$ , (c) ${\textit{VO}}-30^\circ$ and (d) ${\textit{VO}}-45^\circ$ cases.

Figure 16 summarizes the amplitudes and spatial wavenumbers of wall pressure disturbances from DNS, along with N factors and $\alpha _r$ of S-modes from LST. In all cases, three stages are identified: Stage I the leading-edge damping, identified by the steep reduction in both perturbation amplitudes and $\alpha _r$ ; Stage II a non-modal stage, characterized by a secondary decrease of $p^{\prime}_{w, \textit{max}}$ and recovery; Stage III the linear growth stage. The beginnings of non-modal and linear growth are marked by vertical black lines. Prior studies suggest that F-modes are easily excited by vorticity waves (Ma & Zhong Reference Ma and Zhong2005; Balakumar Reference Balakumar2015). However, the leading-edge wavenumbers are incompatible with F-modes, indicating this mechanism does not occur under current flow configurations, where F-modes remain neutral throughout the boundary layer (depicted in figure 6). The decline of $p^{\prime}_{w, max}$ and $\alpha _r$ at the leading-edge peak ( $x \lt 4$ ) is similar to the leading-edge decline observed in acoustic wave cases (figure 9, $x=0$ – $4$ ). This parallel demonstrates that the dissipation may act as the governing mechanism in this stage. Following this dissipating stage, $p^{\prime}_{w, \textit{max}}$ exhibit secondary decrease and recovery (very weak in the ${\textit{VO}}-45^\circ$ case), corresponding to the ‘dead zone’ observed in figure 15. During this regime, $\alpha _r$ surpass LST predictions. Finally, linear growth occurs in Stage III, where both $p^{\prime}_{w, \textit{max}}$ growth and $\alpha _r$ converge with LST predictions. Interestingly, non-modal stages induced by acoustic waves remain nearly constant across different sweep angles, while the non-modal stages persist for a shorter extent under larger sweep angles, implying a stronger receptivity to vorticity waves caused by cross-flow components.

Figure 17.

The instantaneous snapshots of (a,b) acoustic components, (c,d) vortical components and (e,f) thermal components for (a,c,e) ${\textit{VO}}-0^\circ$ and (b,d,f) ${\textit{VO}}-45^\circ$ cases.

Figure 18.

The normalized amplitudes of (a,d) acoustic components, (b,e) vortical components and (c,f) thermal components in (a,c,e) ${\textit{VO}}-0^\circ$ and (b,d,f) ${\textit{VO}}-45^\circ$ cases.

Instantaneous snapshots of MPT components for the ${\textit{VO}}-0^\circ$ and ${\textit{VO}}-45^\circ$ cases are presented in figure 17. Shock waves and boundary layer edges are marked by white solid lines and dashed lines, whereas the vertical black dashed lines refer to the beginning of non-modal growth and linear growth stages. As both the incoming perturbations and S-modes are of a vortical nature, the receptivity process in ${\textit{VO}}$ cases mainly involves the behaviours of $\boldsymbol{m_B^{\prime}}$ . Magnified views of $\boldsymbol{m_B^{\prime}}$ near the leading edge are included to highlight its behaviour. Upstream of shock waves, $\boldsymbol{m_B^{\prime}}$ is uniformly transported by the free stream vorticity waves, while $\boldsymbol{m_A^{\prime}}$ are observed due to the contamination from the shock layer. At the shock layer, $\boldsymbol{m_A^{\prime}}$ undergoes significant generation, alongside weaker $\boldsymbol{m_T^{\prime}}$ generation, while $\boldsymbol{m_B^{\prime}}$ is locally disturbed. Notably, within the shock layer, $\boldsymbol{m_A^{\prime}}$ achieves higher amplitudes compared with its presence in the boundary layer, whereas $\boldsymbol{m_T^{\prime}}$ is less prominent. The transmitted waves remain predominantly vortical in composition. At the boundary layer edge, figures 17(c) and 17(d) reveal a distinctive phenomenon: the vorticity components are effectively blocked from the boundary layer. This observation aligns with the inference of Fedorov & Khokhlov (Reference Fedorov and Khokhlov2001) that the boundary layer effectively swallows vorticity waves. These results provide direct support that the primary source of boundary-layer modes originates from the interactions at the leading edge, with external vorticity waves contributing as a secondary mechanism, transferring limited energy at the boundary-layer edge.

Figure 18 depicts the amplitudes of MPT components at $s=1, 7 \ {\textrm{or}} \ 12.7, 40$ , aiming to capture the leading edge, non-modal stage and linear growth behaviours. The second slices are selected at the minimum of $p^{\prime}_{w, \textit{max}}$ to capture the ‘dead zone’ behaviour. Each amplitude is normalized by the maximum fluctuation momentum density $\boldsymbol{m}^{\prime }$ in the boundary layers. The transmitted wave at $s=1$ is stronger than the boundary layer disturbances. The peak of this slice is plotted in a zoomed-in view for $\boldsymbol{m_B^{\prime}}$ . Unlike ${\textit{AC}}$ cases, the interactions between the vorticity wave, shock wave and boundary layer produce strong acoustic and vortical components, whereas thermal components exhibit lower proportions compared with the linear growth stage. This indicates the leading-edge damping of ${\textit{VO}}$ cases is not induced solely by the acoustic component. In the subsequent non-modal stage, $\boldsymbol{m_A^{\prime}}$ and $\boldsymbol{m_T^{\prime}}$ both exhibit lower proportions compared with the leading-edge profile, forming the secondary decrease of $p^{\prime}_{w, \textit{max}}$ in figure 16. Changes in $\boldsymbol{m_A^{\prime}}$ , $\boldsymbol{m_T^{\prime}}$ and the secondary decrease in case $V0-45^\circ$ are less pronounced. From figures 17(c) and 17(d), $\boldsymbol{m_B^{\prime}}$ seems to undergo modulation at the wavelengths during this stage, whereas $\boldsymbol{m_A^{\prime}}$ and $\boldsymbol{m_T^{\prime}}$ undergo attenuation and recovery. Subsequently, the linear growth phase commences, enabling boundary-layer modes to develop in a manner consistent with LST. The proportions of MPT components become highly similar to ${\textit{AC}}$ cases in figure 12.

The ${\textit{VO}}-0^\circ$ and ${\textit{VO}}-45^\circ$ cases also share the same receptivity mechanisms, with their divergence manifesting in amplitudes of MPT components. The underlying mechanisms of the receptivity pathway can be summarized as: $\boldsymbol{m_B^\prime}$ initially penetrates the boundary layer as a dominant component, then undergo adjustments to fit the linear growth. Here $\boldsymbol{m_A^\prime}$ and $\boldsymbol{m_T^\prime}$ attenuates initially but recovers downstream. These parallels highlight the same physical nature of the first Mack and cross-flow modes.

The receptivity coefficients are then calculated with the definition in Balakumar & King (Reference Balakumar and King2011):

(4.13) \begin{align} C_{VO}=\left (u^{\prime }_{\textit{max} } / |u^{\prime }|_{VO}\right )_\textit{neutral}. \end{align}

These coefficients are summarized in figure 13. All the receptivity coefficients are nearly unity because the neutral points align with the dissipating phase of leading-edge disturbances. This situation renders them inadequate for effectively quantifying the genuine efficiency of vorticity waves in inducing unstable modes.