2.1. Flow configuration

We consider a swept NACA 671–215 (laminar) airfoil with infinite span under a subsonic aircraft-relevant free-stream condition. The basic flow configuration is identical to that used by Kurz & Kloker (Reference Kurz and Kloker2014, Reference Kurz and Kloker2016) for the investigations on surface roughness. The free-stream velocity magnitude is set to $\bar {Q}_\infty =191.8\,\rm{m\,s}^{-1}$ (the overbar denotes dimensional quantities), corresponding to a flight Mach number of $\textit{Ma}_\infty =\bar {Q}_\infty /\bar {a}_\infty =0.65$ , where $\bar {a}_\infty$ is the speed of sound. The orientation of the free-stream velocity vector relative to the swept-wing model is illustrated in figure 1. The sweep angle is defined as $\beta _\measuredangle =\arcsin {(\bar {W}_\infty /\bar {Q}_\infty )}=35^\circ$ , where $\bar {W}_\infty$ is the spanwise free-stream velocity component parallel to the leading edge. The angle of attack, $\alpha _\measuredangle =-6.1^\circ$ , is measured in the plane perpendicular to the leading edge, which corresponds to the angle of attack $\alpha ^*_\measuredangle =-5.0^\circ$ measured with respect to the flight direction. The chord length perpendicular to the leading edge is $\bar {L}=2.0\,\text{m}$ , yielding the Reynolds number $\textit{Re}=\bar {Q}_\infty \bar {\rho }_\infty (\bar {L}/\cos {\beta _\measuredangle })/\bar {\mu }_\infty =10.3\times 10^6$ . The chord length $\bar {L}$ , and the free-stream velocity $\bar {Q}_\infty =191.8\,\rm{m\,s}^{-1}$ , density $\bar {\rho }_\infty$ , temperature $\bar {T}_\infty$ , dynamic viscosity $\bar {\mu }_\infty$ and heat conductivity $\bar {\theta }_\infty$ are used for the non-dimensionalisation of the problem. The most important reference values are summarised in table 1.

Figure 1.

Swept-wing model, orientation of the free-stream velocity and coordinate systems used for simulations and evaluations.

Table 1.

Reference values defining the basic flow set-up.

Table 1. Long description

A table summarizing key reference values for a flow setup involving a swept NACA 671215 airfoil. The table has 13 rows and 4 columns. The columns are labeled Quantity, Symbol, Unit, and Value. The rows list various parameters and their corresponding values. Row 1: Quantity: Angle of attack; Symbol: α∞; Unit: deg; Value: -6.1. Row 2: Quantity: Sweep angle; Symbol: β∞; Unit: deg; Value: 35.0. Row 3: Quantity: Temperature; Symbol: T∞; Unit: K; Value: 216.7. Row 4: Quantity: Density; Symbol: ρ∞; Unit: kg m^-3; Value: 0.31197. Row 5: Quantity: Pressure; Symbol: p∞; Unit: Pa; Value: 19402. Row 6: Quantity: Dynamic viscosity; Symbol: μ∞; Unit: Pa s; Value: 1.4223 x 10^-5. Row 7: Quantity: Velocity magnitude; Symbol: Q∞; Unit: m s^-1; Value: 191.8. Row 8: Quantity: Chordwise velocity; Symbol: U∞; Unit: m s^-1; Value: 157.1. Row 9: Quantity: Spanwise velocity; Symbol: W∞; Unit: m s^-1; Value: 110.0. Row 10: Quantity: Chord length; Symbol: L; Unit: m; Value: 2.0. Row 11: Quantity: Mach number; Symbol: Ma∞; Unit: —; Value: 0.65. Row 12: Quantity: Reynolds number; Symbol: Re; Unit: —; Value: 10.3 x 10^6.

To obtain the initial and boundary conditions for the DNS, a steady solution of the flow around the airfoil is calculated with the Reynolds-Averaged Navier-Stokes (RANS) solver TAU, developed by the German Aerospace Centre (DLR). The transition location is prescribed at 20 % of the chord on both upper and lower sides of the wing. The laminar part of the converged solution is then extracted as initial condition for the subsequent DNS.

2.2. Coordinate systems

The Cartesian global coordinate system $(x, y, z)$ is used for the definition of the airfoil geometry and the simulations. As shown in figure 1, $x$ and $z$ denote the chordwise and spanwise directions, respectively, and the $y$ -direction points upwards. The corresponding velocity components are denoted as $(u, v, w)$ .

For the evaluation of the boundary-layer quantities, we introduce a wall-fitted, curvilinear coordinate system $(\xi , \eta , z)$ with the same origin and $z$ coordinate, while $\xi$ runs perpendicular to the leading edge along the wall and $\eta$ is normal to the wall. The associated velocity components $(u_p,v_p,w_p)$ are aligned with the tangential, wall-normal and spanwise direction, respectively.

The streamline-oriented system $(\xi _s, \eta , z_s)$ is locally aligned with the surface projection of the boundary-layer-edge streamline of the unperturbed laminar base flow. We define the local edge-streamline angle $\phi _e$ as the angle between the wall-parallel component of the edge-velocity vector and the $\xi$ -direction. The corresponding velocity components $(u_s,v_s,w_s)$ can then be expressed using the rotation matrix around the wall-normal axis as

(2.1) \begin{equation} \left ( \begin{array}{c} u_s \\ v_s \\ w_s \\ \end{array} \right ) = \left ( \begin{array}{ccc} \cos \phi _e & 0 & \sin \phi _e \\[3pt] 0 & 1 & 0 \\[3pt] -\sin \phi _e & 0 & \cos \phi _e \\ \end{array} \right ) \left ( \begin{array}{c} u_p \\ v_p \\ w_p \\ \end{array} \right )\!. \end{equation}

Thus, the 3-D boundary-layer flow is locally decomposed into a streamwise, a wall-normal and a CF component.

2.3. Direct numerical simulation

2.3.2. Computational domain and grid generation

The block-structured computational domain consists of two sub-regions, as shown in figure 2. The main block (marked in red) covers the boundary layer in the vicinity of the leading edge and some free-stream extent above. To allow the introduction of controlled disturbances at a location upstream where the free stream is only weakly influenced by the wing, the free-stream block (marked in green) is connected. Note that the outflow region of the free-stream block overlaps with the inflow region of the main block.

Figure 2.

Computational domain for DNS. Contours of the pressure coefficient $c_p= {\bar {p}-\bar {p}_\infty }/{({1}/{2}) \bar {\rho }_\infty \bar {Q}_\infty ^2}$ and streamlines projected in the $xy$ -plane (blue solid lines) resulting from the steady RANS solution are shown.

To obtain the unperturbed DNS base flow, the flow field in the main block is calculated independently. For simulations of the perturbed flow, the disturbance development in the free-stream block is first computed as a preliminary step. The free-stream vortical disturbances are introduced by localised volume forcing, as described in § 2.3.5. The resulting unsteady flow field near the outflow boundary of the free-stream block (shaded region in figure 2) is then extracted as inflow data for the subsequent simulation in the main block, cf. § 2.3.7. Since the free-stream region does not require a very high spatial resolution like the boundary layer, a much larger step size can be chosen for the time integration to reduce the computational cost.

For the grid generation of the main block, a stagnation streamline and two far-field streamlines are extracted from the RANS solution. The far-field boundary consists of an inflow segment perpendicular to the 2-D projection of the stagnation streamline in the $xy$ -plane, and two streamline-shaped free-stream segments, which are smoothly connected by two Bézier curves. The surface points are interpolated by a non-uniform regular B-spline of 20th order with uniform grid spacing along the $\xi$ -direction, except for a refinement area around the stagnation streamlines and the stretched outflow zones. Note that, for the present flow configuration, the attachment line, i.e. the locus of wall points where the stagnation streamlines attach to the surface, does not coincide with the leading edge but is instead located on the wing suction side at $x=0.72$ , as shown in figure 2. To ensure a proper resolution of the boundary layer, the grid in the near-wall region is adjusted to the streamwise boundary-layer thickness $\delta _{99,s}$ resulting from the RANS simulation. For $\eta \leq 2\delta _{99,s}$ , the wall-normal direction is discretised with 75 grid points. The spacing is defined using a geometric series with a common ratio of 1.025. For $\eta \gt 2\delta _{99,s}$ , the near-wall grid is smoothly extended to the far-field boundary. In the vicinity of the stagnation streamline, in particular, the grid lines are aligned with the 2-D projection of the stagnation streamline in the $xy$ -plane. The free-stream block is then generated as a continuous upstream extension of the main block.

The steady RANS solution is interpolated to the DNS grid by a sixth-order, 2-D Lagrangian interpolation. A careful resolution study is conducted to ensure that the investigated free-stream disturbances are resolved with at least 16 points per wavelength along the wall-normal direction before reaching the boundary-layer edge, which is sufficient for the present study.

2.3.5. Controlled free-stream disturbances

Controlled free-stream vortical disturbances are introduced by localised volume forcing slightly downstream of the sponge layer at the inflow of the free-stream block. The derivation of the disturbance input is based on the previous work by Schrader et al. (Reference Schrader, Brandt, Mavriplis and Henningson2010b ), who investigated the receptivity of a 2-D flat plate. For the ease of defining the vortical modes for the 3-D flow configuration in the present study, we introduce a Cartesian coordinate system $(x^*, y^*, z^*)$ which is aligned with the free-stream velocity. The coordinate transform can be expressed in terms of the rotation matrix as

(2.2) \begin{equation} \left ( \begin{array}{c} x \\[3pt] y \\[3pt] z \\ \end{array} \right ) = M \left ( \begin{array}{c} x^* \\[3pt] y^* \\[3pt] z^* \\ \end{array} \right ) \,\,\,\,\,\text{with}\,\,\,\,\, M = \left ( \begin{array}{ccc} \cos \alpha _\measuredangle \cos \beta _\measuredangle & -\sin \alpha _\measuredangle & -\cos \alpha _\measuredangle \sin \beta _\measuredangle \\[3pt] \sin \alpha _\measuredangle \cos \beta _\measuredangle & \cos \alpha _\measuredangle & -\sin \alpha _\measuredangle \sin \beta _\measuredangle \\[3pt] \sin \beta _\measuredangle & 0 & \cos \beta _\measuredangle \\ \end{array} \right )\!, \end{equation}

where $\alpha _\measuredangle$ and $\beta _\measuredangle$ are the angle of attack and the sweep angle, respectively, cf. § 2.1. The disturbance velocity is modelled with Fourier modes of the form

(2.3) \begin{equation} \left ( \begin{array}{c} u'^{*} \\[3pt] v'^{*} \\[3pt] w'^{*} \\ \end{array} \right ) =\left ( \begin{array}{c} \hat {u}'^{*} \\[3pt] \hat {v}'^{*} \\[3pt] \hat {w}'^{*} \\ \end{array} \right ) e^{i(\alpha ^* x^* + \beta ^* y^* + \gamma ^* z^* - \omega t)}, \end{equation}

where the $\hat {\,\,}$ -values denote the complex amplitudes, $\alpha ^*$ , $\beta ^*$ and $\gamma ^*$ are the wavenumbers in the $x^*$ -, $y^*$ - and $z^*$ -directions, respectively, and $\omega$ is the angular frequency. By substituting (2.3) into the incompressible continuity equation, we obtain

(2.4) \begin{equation} \alpha ^{*} \hat {u}'^{*} + \beta ^{*} \hat {v}'^{*} + \gamma ^{*} \hat {w}'^{*}=0. \end{equation}

When the streamwise component $\hat {u}'^{*}$ is set to zero, a vortical mode can be derived, which we define as the ‘A-mode’. The normalised disturbance-velocity amplitude and the corresponding vorticity amplitude read

(2.5) \begin{equation} \text{A-mode:}\,\,\, \left ( \begin{array}{c} \hat {u}'^{*} \\[5pt] \hat {v}'^{*} \\[5pt] \hat {w}'^{*} \\ \end{array} \right ) = \frac {1}{\sqrt {\beta ^{*2}+\gamma ^{*2}}} \left ( \begin{array}{c} 0 \\[5pt] \gamma ^{*} \\[5pt] -\beta ^{*} \\ \end{array} \right )\!,\,\,\,\,\,\, \left ( \begin{array}{c} \hat {\omega }'_{x^*} \\[5pt] \hat {\omega }'_{y^*} \\[5pt] \hat {\omega }'_{z^*} \\ \end{array} \right ) = \frac {i}{\sqrt {\beta ^{*2}+\gamma ^{*2}}} \left ( \begin{array}{c} \beta ^{*2}+\gamma ^{*2} \\[5pt] -\alpha ^{*}\beta ^{*} \\[5pt] -\alpha ^{*}\gamma ^{*} \\ \end{array} \right )\!, \end{equation}

respectively. Similarly, assuming $\hat {v}'^{*}=0$ or $\hat {w}'^{*}=0$ yields

(2.6) \begin{equation} \text{B-mode:}\,\,\, \left ( \begin{array}{c} \hat {u}'^{*} \\[5pt] \hat {v}'^{*} \\[5pt] \hat {w}'^{*} \\ \end{array} \right ) = \frac {1}{\sqrt {\alpha ^{*2}+\gamma ^{*2}}} \left ( \begin{array}{c} -\gamma ^{*} \\[5pt] 0 \\[5pt] \alpha ^{*} \\ \end{array} \right )\!,\,\,\,\,\,\, \left ( \begin{array}{c} \hat {\omega }'_{x^*} \\[5pt] \hat {\omega }'_{y^*} \\[5pt] \hat {\omega }'_{z^*} \\ \end{array} \right ) = \frac {i}{\sqrt {\alpha ^{*2}+\gamma ^{*2}}} \left ( \begin{array}{c} -\alpha ^{*}\beta ^{*} \\ \alpha ^{*2}+\gamma ^{*2} \\ -\beta ^{*}\gamma ^{*} \\ \end{array} \right )\!;\\ \end{equation}

(2.7) \begin{equation} \text{C-mode:}\,\,\, \left ( \begin{array}{c} \hat {u}'^{*} \\[5pt] \hat {v}'^{*} \\[5pt] \hat {w}'^{*} \\ \end{array} \right ) = \frac {1}{\sqrt {\alpha ^{*2}+\beta ^{*2}}} \left ( \begin{array}{c} \beta ^{*} \\[5pt] -\alpha ^{*} \\[5pt] 0 \\ \end{array} \right )\!,\,\,\,\,\,\, \left ( \begin{array}{c} \hat {\omega }'_{x^*} \\[5pt] \hat {\omega }'_{y^*} \\[5pt] \hat {\omega }'_{z^*} \\ \end{array} \right ) = \frac {i}{\sqrt {\alpha ^{*2}+\beta ^{*2}}} \left ( \begin{array}{c} -\alpha ^{*}\gamma ^{*} \\ -\beta ^{*}\gamma ^{*} \\ \alpha ^{*2}+\beta ^{*2} \\ \end{array} \right )\!. \end{equation}

For the disturbance input in the DNS, the corresponding velocity components are transformed back into the global coordinate system as

(2.8) \begin{equation} \left ( \begin{array}{c} u' \\[3pt] v' \\[3pt] w' \\ \end{array} \right ) =\left ( \begin{array}{c} \hat {u}' \\[3pt] \hat {v}' \\[3pt] \hat {w}' \\ \end{array} \right ) e^{i(\alpha x + \beta y + \gamma z - \omega t)},\,\,\,\text{with}\,\,\, \left ( \begin{array}{c} \hat {u}' \\[3pt] \hat {v}' \\[3pt] \hat {w}' \\ \end{array} \right ) = M \left ( \begin{array}{c} \hat {u}'^{*} \\[3pt] \hat {v}'^{*} \\[3pt] \hat {w}'^{*} \\ \end{array} \right )\,\,\,\text{and}\,\,\, \left ( \begin{array}{c} \alpha \\[3pt] \beta \\[3pt] \gamma \\ \end{array} \right ) = M \left ( \begin{array}{c} \alpha ^* \\[3pt] \beta ^* \\[3pt] \gamma ^* \\ \end{array} \right )\!, \end{equation}

where $\alpha$ , $\beta$ and $\gamma$ are the chordwise, vertical and spanwise wavenumbers, respectively.

For the direct excitation of a selected CF instability mode in the swept-wing boundary-layer flow, the angular frequency $\omega$ and the spanwise wavenumber $\gamma$ of the free-stream mode are correspondingly specified, whereas the vertical wavenumber $\beta$ is a free parameter which can be chosen arbitrarily. The streamwise wavenumber $\alpha ^*$ is determined via Taylor’s hypothesis $\omega /\alpha ^*=Q_\infty =1$ . The last parameter defining the Fourier modes, i.e. the chordwise wavenumber $\alpha$ , can thus be found by solving (2.8).

Note that the three different vortical modes derived from (2.4) are linearly dependent, i.e. a mode can be written as a linear combination of the others. In each DNS case, a single vortical mode with a small amplitude is forced near the inflow boundary where the steady base flow is only slightly affected by the wing. However, since the vortical modes defined above are based on the incompressible continuity equation in a uniform free stream, a small adjustment to the underlying flow is expected. Table 2 lists the parameters of the Fourier modes introduced in all DNS cases analysed in this study.

Table 2.

Parameters for the forced free-stream vortical modes in all DNS cases. Here, $|\boldsymbol{k}|$ denotes the wavenumber magnitude. Disturbance-velocity amplitudes are normalised.

Table 2. Long description

A table comparing parameters for forced free-stream vortical modes in DNS cases. The table has 11 rows and 9 columns. The columns are labeled as Case, ω, α, β, γ, |k|, û′, v̂′, and ŵ′. The rows are labeled as A, B, B-β0, B-β1, B-ω+, B-ω+γ+, Combi-B, and Combi-C. The values in the table are as follows: Row A: Case, A; ω, 164.00; α, -1744.76; β, -2684.88; γ, 2356.19; |k|, 3975.48; û′, 0.3361; v̂′, -0.7351; ŵ′, -0.5888. Row B: Case, B; ω, 164.00; α, -1744.76; β, -2684.88; γ, 2356.19; |k|, 3975.48; û′, 0.8469; v̂′, -0.0905; ŵ′, 0.5240. Row B-β0: Case, B-β0; ω, 164.00; α, -1457.78; β, 0.00; γ, 2356.19; |k|, 2770.70; û′, 0.8469; v̂′, -0.0905; ŵ′, 0.5240. Row B-β1: Case, B-β1; ω, 164.00; α, -1171.03; β, 2684.88; γ, 2356.19; |k|, 3759.20; û′, 0.8469; v̂′, -0.0905; ŵ′, 0.5240. Row B-ω+: Case, B-ω+; ω, 328.00; α, -1543.54; β, -2684.88; γ, 2356.19; |k|, 3891.37; û′, 0.8213; v̂′, 0.0279; ŵ′, 0.5698. Row B-ω+γ+: Case, B-ω+γ+; ω, 328.00; α, -2192.20; β, -3579.84; γ, 3141.59; |k|, 5243.15; û′, 0.8628; v̂′, -0.0922; ŵ′, 0.4970. Row Combi-B: Case, Combi-B; ω, 328.00; α, 805.35; β, 0.00; γ, 0.00; |k|, 805.35; û′, 0; v̂′, 0; ŵ′, 1. Row Combi-C: Case, Combi-C; ω, 328.00; α, 805.35; β, 0.00; γ, 0.00; |k|, 805.35; û′, 0; v̂′, 0.9972; ŵ′, -0.0742.

2.3.6. Controlled virtual surface roughness

For investigating the combined receptivity mechanism due to the interaction of free-stream disturbances and surface irregularities, 2-D vortical modes with $\gamma =0$ are chosen as unsteady disturbance input. The spanwise wavenumber of the excited CF mode is then determined by the spectral content of the virtual surface irregularity. To model the effect of a shallow roughness without changing the geometry of the surface mesh, localised steady volume forcing is used in this study. The model is similar to that used in our previous investigation (Guo & Kloker Reference Guo and Kloker2020) on spanwise distributed (T-type) plasma actuators, which were employed in the experiment by Serpieri et al. (Reference Serpieri, Venkata and Kotsonis2017) as virtual roughness elements. See also Dörr & Kloker (Reference Dörr and Kloker2015) for more details on the numerical implementation of the model. Figure 3 shows the distribution of the volume force used in this study and its relative position within the swept-wing boundary layer. Here, the spanwise distributed volume forcing is applied at $x=0.1$ ( $\xi \approx 0.017$ ), slightly downstream of the attachment line. The force maximum is near the wall, oriented in the negative $\xi$ -direction. A weak forcing amplitude is selected such that the locally induced streamwise disturbance velocity reaches approximately 0.1 % of the boundary-layer-edge velocity, similar to the micron-sized roughness elements investigated by Kurz & Kloker (Reference Kurz and Kloker2014) under the same flow configuration. Since the induced disturbance velocity clearly lies within the linear regime, dissipative mechanisms (viscous dissipation and thermal diffusion) are negligibly small, and the perturbation can be regarded as effectively isentropic, i.e. the induced entropy disturbances are negligible.

Figure 3.

(a) Configuration of the virtual roughness element using localised volume forcing. Purple surface: isosurface for $f_{10\,\%}=f_{\textit{max}}/10$ , where $f$ is the magnitude of the volume force. White solid lines: skin-friction lines of the unperturbed base flow. (b) Distribution of the volume force in a cross-cut at $z=0.001$ . Purple lines: $f$ -isolines ( $0.1f_{\textit{max}}$ to $0.9f_{\textit{max}}$ , $\varDelta = 0.2f_{\textit{max}}$ ). Arrows: $(u,v)$ -vector; contour: $w$ . Note that the attachment line is located at $x=0.0072$ ( $\xi =0.0135$ ).

3.1. Boundary-layer characteristics

Boundary-layer parameters downstream of the attachment line on the upper wing side are shown in figure 4. The subscript $s$ denotes values in the streamline-oriented system, cf. § 2.2 and figure 1. The edge velocity $u_{s,e}$ is accelerated due to the favourable pressure gradient. The edge-streamline angle $\phi _e$ continuously decreases downstream of the attachment line at $\xi =0.013$ (corresponding to $x=0.072$ ). Within the boundary layer, the CF component $w_s$ develops, and the normalised value $w_s/u_{s,e}$ reaches its maximum at $\xi \approx 0.04$ and decreases slightly further downstream. The shape parameter $H_{12,s}$ , calculated as the ratio between displacement thickness $\delta _{1,s}$ and momentum thickness $\delta _{2,s}$ , reveals only a very small variation throughout the considered region, indicating near-self-similar velocity profiles in the streamline-oriented system.

Figure 4.

Boundary-layer parameters of the laminar base flow in the streamline-oriented system. The edge-streamline angle $\phi _e$ is given on the right ordinate.

3.2. Stability properties

A spatial linear-stability-theory (LST) solver, see also Schmidt & Rist (Reference Schmidt and Rist2011), is employed to analyse the base flow; surface curvature and non-parallel effects are neglected using the ansatz

(3.1) \begin{equation} \psi '\left (\xi ,\eta ,z,t\right )=\hat {\psi }\left (\eta \right )e^{i\left (\left (\alpha _{p,r}+i\alpha _{p,i}\right ) \xi +\gamma z-\omega t\right )}, \end{equation}

where $\psi$ is a placeholder for the disturbance velocity, density and temperature. For given spanwise wavenumber $\gamma$ and angular frequency $\omega$ , the governing equations are reduced to a generalised eigenvalue problem, in which the complex wavenumber $\alpha _{p,r}+i\alpha _{p,i}$ in the tangential direction $\xi$ is the eigenvalue and the complex 1-D amplitude function $\hat {\psi }(\eta )$ is the eigenfunction.

Figure 5.

Stability diagram for steady CF modes. Contour: spatial amplification rate $-\alpha _{p,i}$ along the tangential coordinate $\xi$ . Black solid lines: amplification factor $N=\int _\xi (-\alpha _{p,i})\text{d} \xi$ . White dashed lines: wave-vector angle $\phi _\alpha$ (spanned by the $\xi$ -axis and the wave vector $(\alpha _{p,r},0,\gamma )$ in the tangential coordinate system).

Figure 6.

Stability diagrams for unsteady CF modes with $\omega =164$ (a) and $\omega =328$ (b). Contour: spatial amplification rate $-\alpha _{p,i}$ along the tangential coordinate $\xi$ . Black solid lines: factor $N=\int _\xi (-\alpha _{p,i})\text{d} \xi$ . White dashed lines: wave-vector angle $\phi _\alpha$ (spanned by the $\xi$ -axis and the wave vector $(\alpha _{p,r},0,\gamma )$ in the tangential coordinate system).

Figure 7.

Top view of typical wave structures associated with different unstable CF modes. (a) Steady mode. (b) Unsteady mode with $\gamma \gt 0$ (travelling in the positive $z$ -direction). (c) Unsteady mode with $\gamma \lt 0$ (travelling in the negative $z$ -direction). Red arrow: wave vector $(\alpha _{p,r},0,\gamma )$ , which corresponds to the phase-speed direction.

The stability diagram for steady disturbances is depicted in figure 5. The highest local amplification rate is found for $\gamma \approx 3300$ at the downstream position $\xi \approx 0.033$ . The $N$ -factor is calculated by integrating $-\alpha _{p,i}$ along the tangential coordinate $\xi$ . The disturbance mode with $\gamma =3142$ is the mode which first attains $N=9$ , therefore regarded as the most dangerous steady mode in triggering transition. The wave-vector angle $\phi _\alpha$ , spanned by the $\xi$ -axis and the wave vector $(\alpha _{p,r},0,\gamma )$ , is marked by the white dashed lines. Since the wave vector is perpendicular to the local wave crest, the wave-vector angle $\phi _\alpha$ characterises the orientation of the corresponding CF modes arising in the boundary layer. To have a better comprehension of the disturbance pattern in the physical space, the wave structure of a typical unstable steady CF mode is illustrated in figure 7(a). In general, the wave vector of a strongly amplified steady mode is nearly perpendicular to the potential-flow streamline of the unperturbed base flow, i.e. the vortex cores of the resulting CFVs are nearly streamwise oriented. The spanwise positions of the steady CFVs are fixed by the initial condition, e.g. the distribution of the surface roughness.

Figure 6 shows the stability diagrams for the unsteady disturbances with $\omega =164$ and $\omega =328$ , which are more relevant for the present study. For $\omega \neq 0$ , the disturbance modes with positive and negative $\gamma$ differ in their travelling direction, cf. figures 7(b) and 7(c). Compared with the steady modes, significantly higher local and integral amplification is achieved by the travelling modes around $\gamma =3142$ , running against the CF. In contrast, the modes with $\gamma \lt 0$ are generally less unstable and hence expected to be less important for the transition process. Moreover, the wave-vector angle $\phi _\alpha$ varies depending on the angular frequency $\omega$ , so that the wave crests of travelling CFVs are typically misaligned with the potential streamlines, as illustrated in figures 7(b) and 7(c).

Figure 8.

Stability diagrams at $\xi =0.03$ (a) and $\xi =0.06$ (b). Contour: spatial amplification rate $-\alpha _{p,i}$ along the tangential coordinate $\xi$ .

In addition, the dependence of the amplification rate $-\alpha _{p,i}$ on the angular frequency $\omega$ and spanwise wavenumber $\gamma$ at two different downstream positions is shown in figure 8. At $\xi =0.03$ , the maximal local amplification is found for $\omega \approx 390$ and $\gamma \approx 3400$ . Travelling modes with positive $\gamma$ are generally more unstable. The unstable $\gamma$ range shrinks with increasing $\omega$ . Farther downstream at $\xi =0.06$ , an overall reduction of the amplification rate is observed for the most unstable modes. A more significant stabilisation is found for the modes with higher $\omega$ . Consequently, the N-factor of the higher-frequency modes are soon overtaken by the lower-frequency ones (not shown). Providing that the free-stream turbulence level is low and the growth of all travelling disturbance modes starts with very low initial amplitudes, the modes with $\omega \gt 400$ are considered less important because they are only strongly amplified for a short distance.

4.1. Streamwise- and cross-wise-vorticity disturbances

First, we compare the capability of different vortical modes with the same frequency and wavenumbers in exciting CFVs in the boundary layer. The low-frequency travelling CF mode with $\omega =164$ and $\gamma =2356$ is chosen as the target mode. It is amplified already at $\xi =0.021$ and attains $N=9$ at $\xi =0.067$ according to LST. A vertical wavenumber of $\beta =-2685$ is selected based on preliminary studies, without specific physical motivation. The effect of varying the vertical wavenumber $\beta$ will be investigated later in § 4.2.

In § 2.3.1, three different vortical disturbance modes have been derived. If the wavenumbers and angular frequency are fixed, the three vortical modes differ in the orientation of their disturbance-velocity amplitude vector. Before the simulation results are presented, some analytical considerations regarding the basic characteristics of these vortical modes are made. First, we note that the spanwise phase speed of the strongly amplified CF modes is typically much smaller than the free-stream velocity, i.e. $\omega /\gamma \ll Q_\infty =1$ , see also figure 6. In the homogeneous free stream, with Taylor’s hypothesis $\omega /\alpha ^* = Q_\infty$ , the streamwise wavenumber $\alpha ^*$ is therefore much smaller than the spanwise wavenumber $\gamma$ . Furthermore, according to (2.8), $\gamma ^*$ can be expressed in terms of $\gamma$ and $\alpha ^*$ as

(4.1) \begin{equation} \gamma ^* = \frac {1}{\cos \beta _\measuredangle }\left ( \gamma - \alpha ^* \sin \beta _\measuredangle \right ). \end{equation}

Since the second term in the brackets is negligible, $\gamma ^*$ is expected to be larger than $\gamma$ and therefore much larger than $\alpha ^*$ . Recalling (2.5), it becomes obvious that the A-mode is characterised by a dominant vorticity component in the streamwise direction while its disturbance-velocity vector is perpendicular to the free-stream direction ( $\hat {u}'^{*}=0$ ). For the B-mode, in contrast, the streamwise disturbance velocity is dominant while the vorticity is nearly perpendicular to the free-stream velocity direction. Since the C-mode can be expressed as a linear combination of the A- and B-modes, which are nearly orthogonal to each other, the detailed analysis of the C-mode is omitted without loss of generality.

Figure 9.

Normalised disturbance-velocity amplitude vector $\hat {\boldsymbol{u}}'=(\hat {u}',\hat {v}',\hat {w}')$ and disturbance-vorticity amplitude vector $\hat {\boldsymbol{\omega }}'=(\hat {\omega }'_x,\hat {\omega }'_{\!y},\hat {\omega }'_z)$ for the free-stream vortical modes with $\omega =164$ , $\gamma =2356$ and $\beta =-2685$ . (a) A-mode (streamwise-vorticity disturbance). (b) B-mode (cross-wise-vorticity disturbance). The orientation of the free-stream velocity vector is displayed for comparison.

For $\omega =164$ , $\gamma =2356$ and $\beta =-2685$ , the normalised amplitude vectors of the corresponding free-stream vortical modes are illustrated in figure 9, cf. table 2 for the values. The basic characteristics of the A-mode (streamwise-vorticity disturbance) and B-mode (cross-wise-vorticity disturbance) discussed before are clearly confirmed.

Figure 10.

Wave structure (snapshot) of the disturbance-vorticity magnitude $\text{sgn}(\omega '_z)|(\omega '_x,\omega '_{\!y},\omega '_z)|$ (negative value when $\omega '_z\lt 0$ ) in a region near the inflow resulting from the B-mode with $\omega =164$ , $\gamma =2356$ and $\beta =-2685$ .

Two DNS cases are now performed, where case A and case B are forced with the A-mode and B-mode at the inflow, respectively. The connected disturbance-velocity vector has a magnitude of 1 % of the base flow for both cases. In figure 10, a snapshot of the wave structure arising from the B-mode near the inflow region of the free-stream block is shown. As discussed in § 2.3.5, since the vortical modes are derived for an incompressible uniform free stream, a minor adjustment occurs after the disturbances are introduced into the DNS. Nevertheless, no significant differences are observed in their spatial wavenumbers or amplitudes compared with the forced values. The wave structure is advected by the mean streamwise velocity towards the wing. Since the forced A-mode has identical wavenumbers, the corresponding wave structure is essentially the same (not shown).

Figure 11.

Downstream development of the streamwise and cross-wise components (denoted by the subscripts $\sigma$ and $\tau$ , respectively) of the disturbance velocity and vorticity for the free-stream vortical modes with $\omega =164$ , $\gamma =2356$ and $\beta =-2685$ . (a) Case A with A-mode. (b) Case B with B-mode. Results are extracted along the 2-D projection of the stagnation streamline in the $xy$ -plane at $t=36T_0$ , $z=0$ .

To investigate the downstream development of the free-stream disturbances, the resulting velocity field is extracted along the 2-D projection of the stagnation/attachment streamline in the $xy$ -plane for both cases. The disturbance-velocity vector is decomposed into a streamwise component $u'_\sigma$ which is aligned with the unperturbed base-flow velocity, and a cross-wise component $u'_\tau$ perpendicular to it. Likewise, the corresponding disturbance-vorticity vector is also decomposed into a streamwise part $\omega '_\sigma$ and a cross-wise part $\omega '_\tau$ . As shown in figure 11, the wavelength of the vortical disturbances is compressed while approaching the attachment line at $x=0.0072$ for both cases due to the deceleration of the base flow. The dominant disturbance component $u'_\tau$ of the A-mode undergoes a continuous damping whereas the dominating component $u'_\sigma$ of the B-mode remains approximately the same amplitude before being strongly attenuated for $x\gt 0$ . The amplitude of the disturbance vorticity is proportional to the (decreasing) amplitude of the disturbance velocity and the (increasing) wavenumber. The dominant streamwise disturbance vorticity of the A-mode is continuously damped, whereas the cross-wise disturbance vorticity of the B-mode is amplified in the free stream before being strongly damped in the region very close to the airfoil.

There are two main effects responsible for the amplitude development of disturbance vorticity in the free stream. The first effect originates from inviscid vortex stretching or compression. When the base flow expands or contracts along the direction of the disturbance-vorticity vector, the vorticity magnitude is correspondingly amplified or attenuated. In particular, the streamwise-vorticity component decreases as the flow approaches the attachment line due to the streamwise deceleration of the base flow, i.e. vortex compression. The cross-wise-vorticity component, however, initially increases because the flow in the vicinity of the attachment line is deflected toward the upper and lower surfaces of the airfoil and accelerated in the cross-wise direction, i.e. vortex stretching. The second effect, in addition, is the viscous damping effect. Because the viscous terms in the governing equations involve second-order spatial derivatives of the velocity field, they typically scale quadratically with the magnitude of the spatial wavenumber for Fourier disturbance modes. Consequently, the viscous damping becomes particularly strong when the wave structure is significantly compressed in the region close to the boundary layer.

Figure 12.

Snapshots of disturbance-vorticity magnitude $\text{sgn}(\omega '_z)|(\omega '_x,\omega '_{\!y},\omega '_z)|$ (negative value when $\omega '_z\lt 0$ ) close to the leading edge, resulting from the free-stream vortical modes with $\omega =164$ , $\gamma =2356$ and $\beta =-2685$ . (a) Case A with A-mode. (b) Case B with B-mode. Ticks along the wing surface indicate the $\xi$ -coordinate.

Snapshots of the wave structures in the vicinity of the swept-wing nose for both cases are shown in figure 12. The orientation of the wave crests in the $xy$ -plane is virtually identical for both cases. However, the overall vortical disturbance amplitude of the A-mode is significantly weaker because of the vortex squeezing effect. Due to the viscous damping effect, both disturbance modes almost die out before reaching the boundary layer.

Figure 13.

Snapshots of streamwise-velocity disturbance $u_s'$ invoked by the free-stream vortical modes with $\omega =164$ , $\gamma =2356$ and $\beta =-2685$ . (a) Case A with A-mode. (b) Case B with B-mode. Wall-parallel coordinate system is used, cf. § 2.2. Not to scale. Dashed lines indicate boundary-layer edge of the unperturbed base flow.

To visualise the boundary-layer response under the influence of free-stream disturbances, snapshots of the streamwise disturbance velocity $u_s'$ within and near the boundary layer are presented in figure 13. In both cases, CF modes are triggered inside the boundary layer and grow in the chordwise direction, while the oncoming free-stream disturbances decay gradually as they propagate downstream. Due to the penetration of the boundary-layer instability modes into the free stream, the obliquely oriented wave structures associated with the free-stream disturbances are eventually replaced by the CF-instability modes, which exhibit vertically (wall-normal) aligned wave structures outside the boundary layer. It is also evident that, because the B-mode retains a higher amplitude when approaching the attachment-line region in case B, the CF mode in the boundary layer is developed earlier than in case A.

Figure 14.

(a) Downstream development of modal streamwise-velocity disturbance amplitudes $\tilde {u}_s'$ with $\omega =164$ and $\gamma =2356$ , resulting from Fourier analysis. The dashed line shows the modal disturbance growth predicted by LST, where the initial amplitude at branch I is chosen arbitrarily, only for comparison with the DNS results shown by the solid lines. (b) Wavenumbers $\alpha _p$ in the $\xi$ -direction of the investigated mode at different positions in the wall-normal direction. The dotted line shows the wavenumber development of the respective eigenmode predicted by LST.

To evaluate the boundary-layer receptivity to the two free-stream disturbance modes, a double Fourier analysis in time and in the spanwise direction is performed, and the streamwise development of the target mode with $\omega = 164$ and $\gamma = 2356$ is presented in figure 14(a). In addition, the LST-predicted amplitude evolution – plotted with an arbitrary initial amplitude at branch I – is superimposed as a dashed line in the same figure for comparison with DNS. Note that, since the attachment line is located at $\xi = 0.013$ , the region with $\xi \lt 0.013$ lies on the opposite side of it. The results indicate that the amplitudes of the target mode in the two DNS cases differ by approximately one order of magnitude near branch I. If receptivity is interpreted as the efficiency of converting far-field free-stream disturbances into boundary-layer instability modes, it becomes evident that the boundary layer is more receptive to the B-mode. This result is consistent with the findings by Tempelmann et al. (Reference Tempelmann, Hanifi and Henningson2012a ), who employed adjoint-based analysis and demonstrated that the optimal free-stream disturbance for exciting steady CF instability modes exhibits a streak-like structure dominated by cross-wise vorticity.

It is worth noting that, apart from the difference in the initial disturbance amplitude level, the downstream evolution in the two cases is almost identical. A common feature observed in both cases is that the targeted mode does not immediately exhibit the high growth rate predicted by LST downstream of branch I. Instead, it experiences a transient adjustment phase, during which its amplitude evolution deviates from the LST prediction, and only farther downstream does its growth rate converge toward that indicated by the dashed LST curve. In fact, the significant deviation of the target mode from the LST behaviour in the transient stage reflects the conversion process through which the free-stream disturbance evolves into a CF instability mode. A key mechanism in this process is the streamwise-wavenumber synchronisation between the free-stream mode and the boundary-layer eigenmode. Figure 14(b) shows the wavenumber $\alpha _p$ of the target mode in the $\xi$ -direction, extracted using double Fourier analysis at two different wall-normal locations inside and outside the boundary layer for both cases. The solid and dashed lines correspond to $\eta = 0.0001$ inside the boundary layer and $\eta = 0.0007$ outside it, cf. figure 13, while the dotted line indicates the wavenumber of the corresponding CF eigenmode predicted by LST. During the initial transient stage, the wavenumber $\alpha _p$ of the free-stream mode outside the boundary layer differs substantially from that of the boundary-layer eigenmode. This discrepancy decreases with increasing $\xi$ , and the two values match at $\xi = 0.037$ . Immediately downstream, the extracted $\alpha _p$ at $\eta = 0.0001$ inside the boundary layer synchronises with the eigenmode value, indicating that the CF instability mode is fully established and remains dominant until the end of the computational domain. Outside the boundary layer, once the CF instability mode overtakes the free-stream disturbance in amplitude, the wavenumber $\alpha _p$ at $\eta = 0.0007$ synchronises with the boundary-layer eigenmode, resulting in the observed jump for both cases.

Although the amplitudes of the forced free-stream modes at the inflow are relatively high (1 % of the free-stream velocity), their magnitudes are substantially reduced before reaching the boundary layer. Additionally, the disturbances excited inside the boundary layer experience a transient stage downstream of branch I, and the exponential growth begins only after the streamwise-wavenumber matching between the free-stream mode and the CF eigenmode. As a result, the target mode amplitude remains below the threshold for inducing secondary instability at $\xi = 0.1$ in either case and therefore does not trigger transition. In contrast, LST predicts that the target mode has already reached $N=15$ at the same location, and even a steady mode with the same spanwise wavenumber would have reached $N=10$ . Hence, the direct receptivity levels exhibited in these two cases are far too low to serve as a realistic mechanism leading to a transition scenario dominated by travelling CFVs.

4.2. Effect of wave-vector orientation

To further examine how different wave orientations of the free-stream vortical disturbances influence the excitation of the CF instability mode, we choose the same travelling mode with $\omega = 164$ and $\gamma = 2356$ as target and vary the vertical wavenumber $\beta$ of the forced free-stream vortical mode at the inflow. Since the previous section has shown that the boundary-layer receptivity to streamwise-vorticity disturbances is much weaker than that to cross-wise-vorticity disturbances, the following studies focus exclusively on the B-mode, which exhibits cross-wise-dominated vorticity and streamwise-dominated velocity perturbations.

Figure 15.

Wave structure (snapshot) of the disturbance-vorticity magnitude $\text{sgn}(\omega '_z)|(\omega '_x,\omega '_{\!y},\omega '_z)|$ (negative value when $\omega '_z\lt 0$ ) in a region near the inflow resulting from B-modes with different wave-vector orientations. (a) Case B- $\beta _1$ with $\omega =164$ , $\gamma =2356$ and $\beta =0$ and (b) case B- $\beta _0$ with $\omega =164$ , $\gamma =2356$ and $\beta =2685$ .

In cases B- $\beta _0$ and B- $\beta _1$ , the free-stream forcing configuration is identical to that of case B, except for the far-field vertical wavenumber, which is prescribed as $\beta = 0$ and $\beta = 2685$ , respectively. The wave structures arising near the inflow region for both cases are shown in figure 15. Note that the wavenumber magnitude of the forced mode in case B- $\beta _1$ is nearly identical to that of case B, whereas that in case B- $\beta _0$ is slightly smaller. Consequently, although the forced disturbance velocity amplitude is the same (1 % of the free-stream velocity) in all three cases, their vorticity amplitudes differ slightly.

Figure 16.

Snapshots of disturbance-vorticity magnitude $\text{sgn}(\omega '_z)|(\omega '_x,\omega '_{\!y},\omega '_z)|$ (negative value when $\omega '_z\lt 0$ ) close to the leading edge. (a) Case B- $\beta _0$ with $\omega =164$ , $\gamma =2356$ and $\beta =0$ (b) case B- $\beta _1$ with $\omega =164$ , $\gamma =2356$ and $\beta =2685$ . Ticks along the wing surface indicate the $\xi$ -coordinate.

As the disturbances propagate toward the wing, the wave orientations differ substantially from those in case B, especially far from the wing, as illustrated in figure 16. Closer to the wing, the wave structures appear similar, as they are highly compressed toward the boundary layer; however, they still remain distinct, which is indicated by the chordwise wavenumbers shown in figure 17(b) (solid lines). Nevertheless, the vorticity magnitudes of the three cases show no significant differences, as vortex stretching and viscous damping outside the boundary layer depend only on the orientation of the vorticity disturbance component and the spatial wavenumbers, respectively, rather than on the wave orientations. The Fourier amplitudes in figure 17(a) show that the boundary-layer disturbance amplitudes are comparable near branch I, whereas the onset of exponential growth varies significantly. As a consequence, the CF instability mode excited in case B- $\beta _1$ is more than one order of magnitude weaker than that in case B at $\xi = 0.1$ .

Figure 17.

(a) Downstream development of modal streamwise-velocity disturbance amplitudes $\tilde {u}_s'$ with $\omega =164$ and $\gamma =2356$ , resulting from Fourier analysis. The dashed line shows the modal disturbance growth predicted by LST, where the initial amplitude at branch I is chosen arbitrarily, only for comparison with the DNS results shown by the solid lines. (b) Wavenumbers $\alpha _p$ in the $\xi$ -direction of the investigated mode at different positions in the wall-normal direction. The dotted line shows the wavenumber development of the respective eigenmode predicted by LST.

In figure 17(b), the disturbance wavenumbers $\alpha _p$ in the $\xi$ -direction extracted at $\eta = 0.0007$ and 0.0001 from cases B and B- $\beta _1$ are compared with the eigenmode wavenumber predicted by LST. Due to the different forced wave orientations for both cases, the free-stream values of $\alpha _p$ at $\eta = 0.0007$ differ considerably, which leads to streamwise-wavenumber matching between the free-stream disturbance (solid lines) and the boundary-layer eigenmode (dotted lines) occurring at different downstream locations. This explains why the onset of the exponential growth in case B- $\beta _1$ occurs noticeably farther downstream than in case B.

Additional parameter studies with more negative values of $\beta$ show that the location of streamwise-wavenumber matching can be shifted further upstream. However, increasing $\left |\beta \right |$ also increases the magnitude of the spatial wavenumber of the forced disturbance mode, which in turn enhances viscous dissipation of the vortical disturbance in the free stream. As a result, such modifications do not lead to improved receptivity of the boundary layer.

4.3. Effect of frequency and spanwise spacing

We continue to use free-stream modes dominated by cross-wise vorticity and streamwise velocity, and with similar wave orientation as that in case B. When $\omega$ and $\gamma$ are varied, the target mode excited inside the boundary layer changes accordingly. Reviewing figures 6 and 8, we observe that the strongly amplified travelling CF modes are concentrated in the ranges $\omega \lt 400$ and $2000\lt \gamma \lt 4000$ ; we therefore progressively increase $\omega$ and $\gamma$ .

Figure 18.

(a) Downstream development of modal streamwise-velocity disturbance amplitudes $\tilde {u}_s'$ with $\omega =164$ or 328 and $\gamma =2356$ , resulting from Fourier analysis. The dashed lines show the modal disturbance growth predicted by LST, where the initial amplitudes at branch I are chosen arbitrarily, only for comparison with the DNS results shown by the solid lines. (b) Wavenumbers $\alpha _p$ in the $\xi$ -direction of the investigated modes at different positions in the wall-normal direction. The dotted lines indicate the wavenumber development of the respective eigenmode predicted by LST.

For case B- $\omega ^+$ , $\omega$ is increased to 328. Figure 18(a) shows that the boundary layer exhibits a higher receptivity to the $\omega = 328$ disturbance. Because the free-stream disturbance mode in case B- $\omega ^+$ experiences somewhat weaker viscous damping before reaching the boundary layer, this leads to a slightly higher initial disturbance amplitude near branch I. In addition, the streamwise-wavenumber matching between the free-stream disturbance and the boundary-layer eigenmode occurs as early as $\xi = 0.025$ , see figure 18(b), the transient region being thus shorter. Consequently, the mode is significantly stronger than that in case B farther downstream and reaches its saturation stage at approximately $\xi = 0.086$ .

Figure 19.

(a) Downstream development of modal streamwise-velocity disturbance amplitudes $\tilde {u}_s'$ with $\omega =328$ and $\gamma =2356$ or 3142, resulting from Fourier analysis. The dashed lines show the modal disturbance growth predicted by LST, where the initial amplitudes at branch I are chosen arbitrarily, only for comparison with the DNS results shown by the solid lines. (b) Wavenumbers $\alpha _p$ in the $\xi$ -direction of the investigated modes at different positions in the wall-normal direction. The dotted lines indicate the wavenumber development of the respective eigenmode predicted by LST.

For case B- $\omega ^+\gamma ^+$ , $\gamma$ is now also increased, to 3142. From figure 19(a) it can be seen that near branch I the disturbance amplitude for $\gamma = 3142$ is more than one order of magnitude lower. This is caused by the stronger viscous damping associated with the higher spatial wavenumber. In addition, streamwise-wavenumber matching with the boundary-layer eigenmode occurs only at approximately $\xi = 0.034$ , shifting the onset of exponential growth farther downstream, and therefore the amplitude level obtained here remains substantially lower than that in case B- $\omega ^+$ .

Extensive parameter studies did not yield any configuration that possesses a noticeably higher boundary-layer receptivity than case B- $\omega ^+$ . Based on our findings, we conclude that increasing $\omega$ while keeping $\gamma$ fixed enhances the boundary-layer response, whereas increasing $\gamma$ has the opposite effect. Furthermore, for cases with $\gamma \lt 0$ – corresponding to CF instability modes whose wave crests travel in the negative $z$ -direction – we never observed streamwise-wavenumber matching between the free-stream mode and the boundary-layer eigenmode, nor any appreciable disturbance amplification.

Even in case B- $\omega ^+$ , which exhibits the strongest boundary-layer response, the initial amplitude measured near branch I is below $10^{-6}$ . Hence, the disturbance requires at least five orders of magnitude ( $N = 11.5$ ) amplification to reach the amplitude level for nonlinear saturation. Recalling that the free-stream mode forced at the inflow already has an amplitude as high as 1 % of the free-stream velocity (corresponding to a much higher Tu level), we conclude that a direct receptivity mechanism driven solely by free-stream vortical disturbances is not likely to produce a realistic transition scenario dominated by travelling CFV modes.