2.2.1. Steady leading-order pressure distribution

There are a number of different approaches ranging in complexity and accuracy to determine the pressure field at the aerofoil surface. We solve the Euler equations directly with the open-source high-order spectral/hp element solver Nektar++ (Cantwell et al. Reference Cantwell2015). This is necessary since, in addition to extracting the pressure distribution, we require the steady inviscid solution in the entire domain as a basic flow for a linearised computation in § 2.3.2; the linear unsteady inviscid solution models the acoustic scattering field due to the incoming acoustic wave interacting with the aerofoil.

For the steady Euler solution, high-order structured meshes are generated with Gmsh (Geuzaine & Remacle Reference Geuzaine and Remacle2009). Nektar++ is parametrised to use a fourth-order discontinuous Galerkin method for the spatial discretisation. The Riemann problem at the interfaces between elements is solved with an ‘exact’ iterative method. A classical fourth-order Runge–Kutta scheme is employed for the temporal discretisation. Computations typically extend up to three convection periods of the far-field flow across the domain, guaranteeing convergence up to the desired precision.

Once a steady converged solution is attained, the pressure coefficient on the aerofoil surface,

(2.3)\begin{equation} c_p(s)=\frac{\bar{p}_e^{*}-\bar{p}_{\infty}}{\dfrac{1}{2} \bar{\rho}_{\infty} \bar{U}_\infty^{2}}, \end{equation}

is the only quantity needed to subsequently perform a steady boundary-layer computation to provide the $\bar {\boldsymbol {q}} (s,n)$ state in (2.2), where $\bar {p}_e^{*}$ is the static pressure at the edge of the boundary layer to the first order of approximation. The remaining flow quantities are recovered through isentropic flow relations.

A criticism of our approach could be that the viscous–inviscid coupling is only considered up to first order, as the effects of the boundary layer on the pressure distribution are not accounted for. Higher-order boundary-layer theory could be implemented for more accurate solutions (Van Dyke Reference Van Dyke1969). However, we only consider high Reynolds number applications, for which high-order corrections of order ${O}(R^{-1/2})$ can be neglected.

2.2.2. Steady boundary-layer field – $\bar {\boldsymbol {q}}$

Let us denote $\bar {\boldsymbol {q}}=[\bar {U},\bar {V},\bar {W},\bar {P}, \bar {\rho }, \bar {T}]^{\text {T}}$ the steady base flow quantities. We restrict our attention to two-dimensional flows, i.e. $\bar {W}=0$. It is convenient to introduce the generalised Howarth–Dorodnitsyn transformation (Moore Reference Moore1951; Stewartson Reference Stewartson1951) in order to eliminate the continuity equation from the system of equations. We define the change of variable

(2.4)\begin{equation} {\eta}=\left( \frac{\bar{U}_e^{*}}{\bar{\mu}_e^{*} \bar{\rho}_e^{*} s^{*}} \right)^{1/2} \int_0^{n^{*}} \bar{\rho}^{*}\,{\textrm{d}}s, \end{equation}

and the streamfunction

(2.5)\begin{equation} {\psi}=\left(\bar{\rho}_e^{*} \bar{\mu}_e^{*} \bar{U}_e^{*} s^{*} \right)^{1/2} {F}({\eta}, s^{*}), \end{equation}

with

(2.6a,b)\begin{equation} F_{\eta} = \frac{{\bar{U}}^{*}}{\bar{U}_e^{*}},\quad S=\frac{{\bar{T}}^{*}}{\bar{T}_e^{*}}, \end{equation}

where the subscript ‘$e$’ denotes boundary-layer edge quantities. For the purposes of this subsection we consider that all flow and material quantities are made non-dimensional using local boundary-layer edge quantities instead of the far-field quantities used in (2.1).

The steady boundary-layer equations (see, for example, Schlichting & Gersten Reference Schlichting and Gersten1960) are then simplified using (2.4)–(2.6a,b), yielding

(2.7a)\begin{gather} \frac{\partial \left( {m_0} {F}_{{\eta} {\eta}} \right)}{\partial {\eta}} + m_3 {F}_{{\eta} {\eta}} {F} + m_1 \left( S - {F}_{{\eta}}^{2} \right) = s \left( {F}_{{\eta}} {F}_{{\eta} s} - {F}_{{\eta} {\eta}} {F}_{s} \right), \end{gather}

(2.7b)\begin{gather}\frac{1}{\sigma} \frac{\partial \left({m_0} S_{{\eta}} \right)}{\partial {\eta}} + m_3 F S_{{\eta}} + {m_0} \left( \gamma-1 \right) M_e^{2} F_{{\eta} {\eta}}^{2} = s \left( F_{{\eta}} S_s - S_{{\eta}} F_s \right), \end{gather}

where,

(2.8)\begin{equation} \left.\begin{gathered} {m_0}={\bar{\mu}} {\bar{\rho}},\quad m_1=\frac{s}{\bar{U}_e^{*}} \frac{d \bar{U}_e^{*}}{d s},\quad m_2=\frac{s}{\bar{\mu}_e^{*} \bar{\rho}_e^{*}} \frac{{\textrm{d}} \left( \bar{\mu}_e^{*} \bar{\rho}_e^{*} \right)}{{\textrm{d}}s}, \\ m_3=\frac{1}{2} \left( 1+m_1+m_2 \right),\quad M_e^{2}= \frac{\bar{U}_e^{*^{2}}}{\gamma \mathcal{R} \bar{T}_e^{*}}. \end{gathered}\right\} \end{equation}

The subscripts $\eta$ and $s$ indicate partial derivatives with respect to these variables. The Prandtl number, the specific heat ratio and the ideal gas constant are denoted $\sigma$, $\gamma$ and $\mathcal {R}$ respectively. Note that the curvature of the body has no explicit influence on the boundary layer to leading order. Curvature-related corrections appear in high-order boundary-layer theory only (Schlichting & Gersten Reference Schlichting and Gersten1960).

The usual no-slip and isothermal or adiabatic boundary conditions are used. Equation (2.7) is solved to second-order accuracy, using upwind finite differences in the streamwise direction and the fully implicit Keller-box method in the wall-normal direction. Newton iterations are used to arrive at a converged solution of the nonlinear governing equations (see more detail in § 2.5 of Raposo Reference Raposo2020). The marching procedure is stopped when the solver fails to converge due to the Goldstein singularity (Goldstein Reference Goldstein1948).

2.3.1. Free-stream acoustic-wave solution

Our analysis of the acoustic field starts in the far-field region, where the uniform basic flow is unaffected by the presence of the aerofoil. Let us consider a small-amplitude acoustic wave travelling in the free stream with the unperturbed uniform steady flow,

(2.9)\begin{equation} \boldsymbol{q}_\infty(x,y,t)=\bar{\boldsymbol{q}}_\infty+ \varepsilon_a \boldsymbol{q}_{a,\infty} \varUpsilon, \end{equation}

where $\varUpsilon = \exp \{\textrm {i} \alpha _{a} (- c_a t + x+ \lambda _1 y ) \}$ and $\bar {\boldsymbol {q}}_\infty =[ 1, 0, 0,\bar {p}_\infty / \bar {\rho }_\infty \bar {U}^{2}_\infty ,1,1 ]$. This implies $\omega =\alpha _{a} c_a$, where $\omega$, $\alpha _a$ and $c_a$ denote the angular frequency, streamwise wavenumber and phase speed of the wave solution. The direction of propagation of the acoustic wave is determined by parameter $\lambda _1$. Its relationship with the more physical angle of incidence $\varTheta _i$ (see definition in § 2.1) for subsonic flows is

(2.10)\begin{equation} \lambda_1=\text{tan} \left( {\rm \pi}- \varTheta_i - \alpha \right). \end{equation}

The acoustic-wave dispersion relation is known to be

(2.11)\begin{equation} c_a=1 \pm \frac{\sqrt{1+\lambda_1^{2}}}{M_\infty}, \end{equation}

whereas the acoustic-wave amplitude is

(2.12a–d)\begin{align} u_{a,\infty}=\frac{p_{a,\infty}}{c_a-1},\quad v_{a,\infty}=\lambda_1 u_{a,\infty},\quad \theta_{a,\infty}=(\gamma-1) M_{\infty}^{2} p_{a,\infty},\quad \rho_{a,\infty}=M_{\infty}^{2} p_{a,\infty}. \end{align}

The far-field Mach number $M_\infty =\bar {U}_\infty /\sqrt {\gamma \mathcal {R} \bar {T}_\infty }$ was introduced, as well as $\theta _a$ and $\rho _a$, the temperature and density perturbations. The acoustic pressure perturbation $p_{a,\infty }=1/M_\infty$ is constant with respect to the angle of incidence and corresponds to the downstream-travelling wave solution $u_{a,\infty }(\varTheta _i=-\alpha )=1$ (see Raposo et al. (Reference Raposo, Mughal and Ashworth2019) for more details).

2.3.2. Inviscid region

As the plane acoustic wave arriving from the far field approaches the aerofoil, the non-uniform steady basic flow modifies the acoustic wave solution. With the exception of the viscous wall layer, this flow is inviscid to leading order of approximation and is therefore adequately modelled by the linearised Euler equations for small amplitudes. Similar to the procedure adopted for the basic flow, the goal is to determine the unsteady pressure perturbation over the aerofoil surface which drives the unsteady acoustic boundary layer.

The present work uses a numerical tool developed by Bensalah (Reference Bensalah2018) to model acoustic-wave propagation in inviscid flows. Herein we will only describe the most salient points of this approach. The total flow is considered inviscid, irrotational and homentropic. Let us further consider a small-amplitude acoustic wave travelling with the potential steady basic flow,

(2.13)\begin{equation} \boldsymbol{q}(x,y,t)=\boldsymbol{q}_i(x,y)+ \varepsilon_a \boldsymbol{q}_{h}(x,y) \exp\{- \textrm{i} \omega t \}. \end{equation}

The subscript $i$ denotes the known inviscid basic flow resulting from the computation described in § 2.2.1 and the subscript $h$ denotes the unknown acoustic-wave quantities, strictly a function of the coordinates $(x,y)$. Under these conditions the perturbation flow can be described by a potential function $\varphi$ according to

(2.14)\begin{equation} \boldsymbol{v}_h=\boldsymbol{\nabla} \varphi,\end{equation}

where $\boldsymbol {v}_h=[ u_h,v_h,w_h ]^{\text {T}}$. These hypotheses, alongside a perfect gas law, allow for a simplification of the linearised Euler equations into a single scalar partial differential equation

(2.15)\begin{equation} D_t \left(c_i^{{-}2} D_t \varphi \right) - \rho_i^{{-}1} \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho_i \boldsymbol{\nabla} \varphi \right) = 0,\end{equation}

where $c_i=\sqrt {\gamma \mathcal {R} T_i}$ is the local speed of sound and $D_t=-\textrm {i}\omega + u_i \partial /\partial x+ v_i \partial /\partial y$. A detailed derivation of this equation can be found in Howe (Reference Howe1998). On solution of (2.15), we can recover the remaining physical flow variables with the scalar equations

(2.16a)\begin{gather} p_h={-}\rho_i D_t \varphi, \end{gather}

(2.16b)\begin{gather}\theta_h={-} \left( \gamma-1 \right) M_{\infty}^{2} D_t \varphi, \end{gather}

(2.16c)\begin{gather}\rho_h={-} \rho_i c_i^{{-}2} D_t \varphi. \end{gather}

The flow respects the impermeability boundary condition at the surface of the aerofoil

(2.17)\begin{equation} \boldsymbol{\nabla} \varphi \cdot \boldsymbol{n} = 0, \end{equation}

where $\boldsymbol {n}$ is the unit normal vector to the surface. The far-field boundary condition is not straightforward because while we know the form and amplitude of the incident acoustic wave, the reflected outgoing waves must be determined as part of the solution. The classical approach of scattering theory is to decompose the total flow as

(2.18)\begin{equation} \varphi= \varphi_r + \varphi_a, \end{equation}

where $\varphi _r$ is the unknown reflected and scattered acoustic field, and $\varphi _a$ is the known far-field acoustic-wave solution determined in § 2.3.1

(2.19)\begin{equation} \varphi_a={-} \frac{\textrm{i}p_{a,\infty} }{\omega-\alpha_a } \exp \{\textrm{i}\left( \alpha_a x + k_a y \right) \}, \end{equation}

where $k_a=\lambda _1 \alpha _a$. Substituting (2.18) into (2.15) yields

(2.20)\begin{equation} D_t \left(c_i^{{-}2} D_t \varphi_r \right) - \rho_i^{{-}1} \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho_i \boldsymbol{\nabla} \varphi_r \right) ={-}D_t \left(c_i^{{-}2} D_t \varphi_a \right) + \rho_i^{{-}1} \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho_i \boldsymbol{\nabla} \varphi_a \right), \end{equation}

whereas the boundary condition (2.17) becomes

(2.21)\begin{equation} \boldsymbol{\nabla} \varphi_r \cdot \boldsymbol{n} ={-} \boldsymbol{\nabla} \varphi_a \cdot \boldsymbol{n} \quad \text{at the rigid surface}. \end{equation}

In flow regions where the basic flow is uniform, the decomposition (2.18) corresponds exactly to the incident and reflected waves. In such cases, the incident acoustic wave $\varphi _a$ satisfies the convected Helmholtz equation and therefore (2.20) is reduced to

(2.22)\begin{equation} D_t \left(c_i^{{-}2} D_t \varphi_r \right) - \rho_i^{{-}1} \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho_i \boldsymbol{\nabla} \varphi_r \right) = 0. \end{equation}

However, as we approach the aerofoil, the term $\varphi _r$ models not only the reflected and scattered wave solution but also the distortion of the incident acoustic wave caused by the change in the basic flow. In this case, the known right-hand side of (2.20) is non-zero and forces the appearance of an acoustic perturbation. The key advantage of introducing this decomposition is that it transforms the incident inhomogeneous boundary conditions into a forcing term of the governing equation and an inhomogeneous rigid surface boundary condition. Consequently, it becomes easier to impose boundary conditions in the far field, where a perfectly matched layer (PML) technique (Bécache, Dhia & Legendre Reference Bécache, Dhia and Legendre2004) is implemented to completely dampen the outgoing acoustic wave and avoid spurious reflections inside the computational domain. In this case, we can approximate the far-field boundary condition after the PML by

(2.23)\begin{equation} \varphi_r = 0. \end{equation}

(2.20) together with boundary conditions (2.21) and (2.23) are solved with a classic first-order finite-element code in triangular unstructured meshes generated by Gmsh (Geuzaine & Remacle Reference Geuzaine and Remacle2009). The reader is directed to Bensalah (Reference Bensalah2018) for details of the implementation. A diagram of the computational domain and a representation of the PMLs are shown in figure 4. The subscripts indicate the direction of propagation along which the sponge layer acts to dampen the solution.

Figure 4. Inviscid acoustic propagation problem – schematic diagram of the computational domain.

2.3.3. Linearised unsteady boundary-layer equations – $\boldsymbol {q}_a$

Viscous effects have thus far been neglected both in the basic flow and in the acoustic field. To correct for this, and to respect the no-slip boundary condition, we use the unsteady boundary-layer equations to model the acoustic boundary layer. Consider a perturbation superimposed on the steady boundary-layer base flow quantities,

(2.24)\begin{equation} \boldsymbol{q}(s,n,t)=\bar{\boldsymbol{q}}(s,n)+ \varepsilon_a \boldsymbol{q}_{a}(s,n) \exp\{- \textrm{i} \omega t \}. \end{equation}

The acoustic perturbations $\boldsymbol {q}_{a}$ are functions of $(s,n)$, apart from the pressure perturbation considered not to vary in the wall-normal direction. Further consider the unsteady compressible boundary-layer equations

(2.25a)\begin{gather} \rho_{t}+(\rho u)_{s} + (\rho v)_{n} =0, \end{gather}

(2.25b)\begin{gather}\rho \left( u_{t} + u u_{s} + v u_{n} \right) ={-} p_{s} + \frac{1}{R} \left(\mu u_{n} \right)_{n}, \end{gather}

(2.25c)\begin{gather}p_{n}=0, \end{gather}

(2.25d)\begin{gather}\sigma \rho \left( T_{t}+ u T_{s} + v T_{n} \right) = \frac{1}{R} \left(\mu T_{n} \right)_{n} + \varGamma \left( p_{t} + u p_{s} \right) + \frac{\varGamma \mu}{R} u_n^{2}, \end{gather}

which are made non-dimensional according to (2.1), where $R=\bar {U}_\infty c_n / \bar {\nu }_\infty$ is the global Reynolds number and $\varGamma =(\gamma -1) M_\infty ^{2} \sigma$. We next substitute the flow-field expansions (2.24) in the above equations, linearise around the mean flow and make a change of variable according to (2.4), yielding

(2.26a)\begin{align} & \eta_n \bar{\rho} \frac{\partial v_a}{\partial \eta} + \left( \eta_s \bar{\rho}_\eta + \bar{\rho}_s \right) u_a + c_3 \left( 2 c_0 c_5 + \bar{U} (2 c_6 - c_4) -c_1 - \bar{U}_s - \textrm{i} c_2 \right) \theta_a \nonumber\\ &\quad + \eta_s \bar{\rho} \frac{\partial u_a}{\partial \eta}+ \bar{\rho} \frac{\partial u_a}{\partial s} - c_0 c_3 \frac{\partial \theta_a}{\partial \eta} + \eta_n \bar{\rho}_\eta v_a - c_3 \bar{U} \frac{\partial \theta_a}{\partial s} ={-}c_7 \bar{U} \frac{\partial p_a}{\partial s} \nonumber\\ &\quad - c_7\left( c_1 + \bar{U}_s + \textrm{i} c_2 -c_0 c_5 - c_6\bar{U} \right) p_a, \end{align}

(2.26b)\begin{align} & \left[ R^{{-}1} \bar{\mu}_{\bar{T}} \bar{U}_{\eta \eta} \eta_n^{2} + R^{{-}1} \left( \bar{\mu}_{\bar{T}} \eta_{nn} + \bar{\mu}_{\bar{T} \bar{T}} \bar{T}_\eta \eta_n^{2} \right) \bar{U}_\eta + c_3 c_8 \right] \theta_a \nonumber\\ &\qquad + \frac{\bar{\mu} \eta_n^{2}}{R} \frac{\partial^{2} u_a}{\partial \eta^{2}} + R^{{-}1} \bar{\mu}_{\bar{T}} \eta_n^{2} \bar{U}_\eta \frac{\partial \theta_a}{\partial \eta}+ \left( R^{{-}1} \bar{\mu} \eta_{nn} + R^{{-}1} \bar{\mu}_{\bar{T}} \bar{T}_\eta \eta_n^{2} - \bar{\rho} c_0 \right) \frac{\partial u_a}{\partial \eta} \nonumber\\ &\qquad - \left[ \bar{\rho}\left( \eta_s \bar{U}_\eta + \bar{U}_s \right) + \textrm{i} c_2 \bar{\rho} \right] u_a - \eta_n \bar{\rho} \bar{U}_\eta v_a - \bar{\rho} \bar{U} \frac{\partial u_a}{\partial s} \nonumber\\ &\quad = \frac{\partial p_a}{\partial s} + c_7 c_8 p_a, \end{align}

(2.26c)\begin{align} & 2 \eta_n^{2} R^{{-}1} \bar{\mu} \varGamma \bar{U}_\eta \frac{\partial u_a}{\partial \eta} - \left[ \sigma \bar{\rho} \left( \eta_s \bar{T}_\eta + \bar{T}_s \right) - \varGamma \bar{P}_s \right] u_a - \eta_n \sigma \bar{\rho} \bar{T}_\eta v_a + \frac{\bar{\mu} \eta_n^{2}}{R} \frac{\partial^{2} \theta_a}{\partial \eta^{2}} \nonumber\\ &\qquad - \sigma \bar{\rho} \bar{U}\frac{\partial \theta_a}{\partial s} + \left( 2 \eta_n^{2} R^{{-}1} \bar{\mu}_{\bar{T}} \bar{T}_\eta + \eta_{nn} R^{{-}1} \bar{\mu} - \sigma c_0 \bar{\rho} \right) \frac{\partial \theta_a}{\partial \eta} \nonumber\\ &\qquad + \left[ \eta_n^{2} R^{{-}1}\bar{\mu}_{\bar{T}} \bar{T}_{\eta \eta} +\eta_{nn} R^{{-}1} \bar{\mu}_{\bar{T}} \bar{T}_\eta + \eta_n^{2} R^{{-}1} \bar{\mu}_{\bar{T} \bar{T}} \bar{T}_{\eta}^{2} \right. \nonumber\\ &\qquad \left. + \sigma c_3 \left( c_0 \bar{T}_\eta + \bar{U} \bar{T}_s \right) + \eta_n^{2} R^{{-}1} \varGamma \bar{\mu}_{\bar{T}} \bar{U}_\eta^{2} - \textrm{i} c_2 \sigma \bar{\rho} \right]\theta_a \nonumber\\ &\quad = \left[ \sigma c_7 \left( c_0 \bar{T}_\eta + \bar{U} \bar{T}_s \right) - \textrm{i} c_2 \varGamma \right] p_a - \varGamma \bar{U} \frac{\partial p_a}{\partial s}, \end{align}

where

(2.27)\begin{equation} \left.\begin{gathered} c_0=\eta_s \bar{U} + \eta_n \bar{V},\quad c_1=\eta_s \bar{U}_\eta + \eta_n \bar{V}_\eta,\quad c_2={-} \omega,\quad c_3=\frac{\bar{\rho}}{\bar{T}},\quad c_4=\frac{\bar{P}_s}{\bar{P}},\\ c_5= \frac{\bar{T}_\eta}{\bar{T}},\quad c_6=\frac{\bar{T}_s}{\bar{T}}, c_7=\frac{\bar{\rho}}{\bar{P}},\quad c_8=c_0 \bar{U}_\eta + \bar{U} \bar{U}_s. \end{gathered}\right\} \end{equation}

The linearised ideal gas law closes the system of equations

(2.28)\begin{equation} \gamma M_{\infty}^{2} p_a= \bar{\rho} \theta_a + \bar{T} \rho_a. \end{equation}

The pressure perturbation $p_a(s)$ driving the unsteady motion within the boundary layer (right-hand sides of (2.26a)–(2.26c)) is prescribed by the outer unsteady inviscid flow examined in § 2.3.2 – $p_h(x,y)$ evaluated at the aerofoil surface from (2.16). The corresponding flow velocity and temperature determine the boundary conditions at the edge of the boundary layer

(2.29a,b)\begin{equation} u_a(\eta \rightarrow \infty)=\sqrt{u_h^{2}+v_h^{2}},\quad \theta_a(\eta \rightarrow \infty)=\theta_{h}, \end{equation}

where the outer layer quantities denoted by the subscript $h$ are evaluated at the aerofoil surface (matching condition). At the wall we impose no-slip boundary conditions

(2.30a–c)\begin{equation} u_a(s,\eta=0)=0,\quad v_a(s,\eta=0)=0,\quad \theta_a(s,\eta=0)=0. \end{equation}

The thermal fluctuation $\theta _a$ is also assumed to have zero fluctuation at the surface. This is based on the assumption that the perturbations oscillate at high temporal frequencies. These oscillations are too fast for the thermal dynamics to react and the wall temperature to adjust (Mack Reference Mack1984).

The numerical methods to obtain a solution are very similar to those used in § 2.2.2 for the steady boundary-layer field. In this case the equations are linear and therefore Newton iterations are not required. A solution can be obtained directly once the steady boundary-layer field solution converges at each streamwise location during the parabolic numerical marching procedure (see more detail in § 2.5 of Raposo Reference Raposo2020).

3.5.1. Effects of localised roughness position

In figure 15(a) we present the variation of the equivalent receptivity amplitude with the position of a Gaussian-shaped roughness element $s_b$ for a fixed width $\varDelta ^{*}/c_n=7.53 \times 10^{-3}$. Two sets of results were obtained with the adjoint approach, one for each of the meshes used in the inviscid acoustic propagation calculations shown in figure 12. Acoustic receptivity is known to be very sensitive to the acoustic-wave boundary-layer signature, which in turn depends on the boundary-layer edge conditions obtained from these computations. Performing this comparison thus confirms that the inviscid acoustic propagation results are grid independent. A third set of results obtained with the direct HLNS approach is superimposed in figure 15(a). The excellent agreement with the adjoint approach supports the correctness of the implementation of both methods. In figure 15(b) we present the corresponding efficiency function using two different grids for the AHLNS computations to verify grid independence.

Figure 15. Receptivity amplitude variation with the position of the localised Gaussian-shaped roughness element for $\mathcal {S}=46.16$, $\varDelta ^{*}/c_n=7.53 \times 10^{-3}$, $\varTheta _i=45^{\circ }$ and $\alpha =0^{\circ }$. First branch of neutral stability (B-I) marked by a red cross. (a) Boundary-layer instability amplitude at the first branch of neutral stability. Comparison between the direct and adjoint approach. (b) Efficiency function – HLNS grid refinement study.

We note from figure 15(a) that, unlike with zero-pressure-gradient boundary layers developing over a flat plate, receptivity is not maximal in the vicinity of the first branch of neutral stability; nor does it decay monotonically when the roughness element is located upstream or downstream of this point. Instead, non-trivial variations appear which we mostly attribute to equally non-trivial behaviour in the acoustic-wave boundary-layer signature (see figure 12b).

In figure 16 we show how the results presented in figure 15(a) change when varying the acoustic-wave angle of incidence as well as the angle of attack at a constant frequency. Firstly, note how the boundary layer is significantly more receptive to upstream-travelling waves than to downstream-travelling waves at all angles of attack. This trend is similar to what has been observed for flat plate geometries using a similar modelling approach, although in the present case the maximum does not occur for $\varTheta _i=180^{\circ }$ (see Raposo et al. Reference Raposo, Mughal and Ashworth2019). This is largely due to the inner layer of the acoustic boundary layer, also known as the Stokes layer, having larger streamwise velocity oscillations when excited by upstream-travelling waves. The physical mechanism is most easily understood when considering the asymptotic layers of the acoustic field. Similarly to a steady boundary layer, the acoustic boundary layer is driven by an externally imposed pressure gradient. From (2.26b), it can be seen that the forcing of the streamwise momentum equation is given by

(3.12)\begin{equation} \frac{\partial p_a}{\partial s} + c_7 c_8 p_a. \end{equation}

If we neglect the plane acoustic-wave distortion by the non-uniform mean flow, it can be concluded that the streamwise pressure gradient $\partial p_a/\partial s$ is larger for upstream-travelling waves – this is readily seen from the far-field acoustic-wave solution presented in § 2.3.1. Since acoustic receptivity is linear with respect to the acoustic field, we can thus expect upstream-travelling acoustic waves to generate T–S waves with larger initial amplitudes.

Figure 16. Equivalent receptivity amplitude variation with the position of the Gaussian-shaped roughness element. Comparison for different acoustic wave angles of incidence at three angles of attack ($\mathcal {S}=46.16$, $\varDelta ^{*}/c_n=7.53 \times 10^{-3}$); (a) $\alpha =-2^{\circ }$, (b) $\alpha =0^{\circ }$ and (c) $\alpha =2^{\circ }$.

However, it was found in flat plate studies in Raposo (Reference Raposo2020) that viscosity plays a significant role in the acoustic reflection and hence in receptivity to highly oblique upstream-travelling waves – not accounted for in this paper, since we address the purely inviscid scattering problem presently. The LUBLE become invalid when transverse pressure variations become significant. Scale analysis of the convective terms of the $x$-momentum equation indicates that this occurs when the streamwise wavelength of the acoustic wave is comparable to the boundary-layer thickness, i.e. $\delta ^{*}={O}(1/\alpha _a^{*})$. This condition is verified for high frequencies and Mach numbers, and for highly oblique upstream-travelling waves. The exact behaviour in these conditions remains an open question even in a flat plate since Raposo (Reference Raposo2020) showed that alternative theories based on the LSE fail in such conditions too.

From 16 we note how the location of the first branch of neutral stability, $s_b=\{ 0.363,0.285,0.196\}$ for $\alpha =\{-2,0,2\}^{\circ }$ respectively, continues to be a reliable indicator of where receptivity is maximal for some angles of incidence, but not for others. Receptivity is typically maximal just ahead of the first branch of neutral stability because this is the location where the resonance conditions with external forcing are optimally met. This means that as the angle of attack increases, the most receptive region of the boundary layer moves upstream towards the leading edge.

Higher angles of attack are associated with stronger adverse pressure gradients in the region where the boundary layer is most receptive. For reference, the suction peak in the upper surface of the aerofoil occurs at $x^{*}/c_n=\{0.23,0.12,0.036\}$ for $\alpha =\{-2,0,2\}^{\circ }$, respectively. Therefore, it can be seen from figure 16 that the boundary layer is most receptive in regions of adverse pressure gradient for the particular frequency considered. Nonetheless, there is no clear trend correlating the angle of attack and the receptivity amplitudes. Relative differences can, however, be explained qualitatively by examining the inviscid acoustic wave's slip velocity on the aerofoil upper surface shown in figure 17 for two angles of incidence of the acoustic wave. Larger amplitudes of the Stokes layer correlate with higher levels of receptivity, thus explaining the results in figure 16.

Figure 17. Inviscid acoustic-wave streamwise velocity at the upper surface of the aerofoil. Comparison for different angles of attack at two angles of incidence of the acoustic wave ($\mathcal {S}=46.16$); (a) $\varTheta _i=25^{\circ }$ and (b) $\varTheta _i=155^{\circ }$.

Thus far we have presented and discussed results for a single frequency. In figure 18 we extend the study to higher frequencies which also correspond to higher $N$-factors. The associated boundary-layer instabilities are thus likely to play a more prominent role in transition in this configuration. Crucially, we observe that the most receptive locations move upstream with an increase in frequency, similarly to the first branch of neutral stability. This is accompanied by an apparent decrease in receptivity amplitudes. In the next section we will see that this can be attributed to the constant width of the roughness element which does not resonate optimally for all frequencies.

Figure 18. Equivalent receptivity amplitude variation with the position of the Gaussian-shaped roughness element. Comparison for different acoustic wave angles of incidence at three frequencies ($\alpha =0^{\circ }$, $\varDelta ^{*}/c_n=7.53 \times 10^{-3}$); (a) $\mathcal {S}=46.16$, (b) $\mathcal {S}=69.24$ and (c) $\mathcal {S}=115.4$.

3.5.2. Effects of localised roughness width

In figure 19 we show the variation of the T–S wave amplitude when changing the width of the Gaussian bump for several fixed locations $s_b$. Parallel flow theory predicts that receptivity amplitudes vary in proportion to the Fourier transform of the surface roughness geometry, in this case

(3.13)\begin{equation} A_0 \sim \varDelta \exp\left( - 2 {\rm \pi}^{2} k^{2} \varDelta^{2} \right), \end{equation}

where $k$ is the wavenumber. According to parallel flow theory (Choudhari Reference Choudhari1994a), the resonant wavenumber of the roughness field is given by $k=1/\lambda _{{TS}}-1/\lambda _{{ac}}$, where $\lambda _{{TS}}$ is the T–S wavelength and $\lambda _{{ac}}$ is the acoustic-wave streamwise wavelength. The optimal width is thus

(3.14)\begin{equation} \varDelta^{*}/c_n=\frac{\lambda_{{TS}}^{*}}{2 {\rm \pi}c_n} \frac{1}{1- \dfrac{\lambda_{{TS}}^{*}}{\lambda_{{ac}}^{*}}}, \end{equation}

where $\lambda _{{ac}}^{*}$ can be estimated as

(3.15)\begin{equation} \lambda_{{ac}}^{*}=\left. \frac{2 {\rm \pi}|u_a^{*}|}{\dfrac{\partial |u_a^{*} |}{\partial s^{*}}} \right|_{\left( s_b, n \rightarrow \infty \right)}. \end{equation}

This result is a necessary resonant condition for receptivity stemming from (2.31) under a parallel flow assumption. For example, for $\alpha =0^{\circ }$ and $s_b=\{0.15,0.25\}$, parallel flow theory predicts receptivity maxima at $\varDelta ^{*}/c_n=\{8.23,8.64\} \times 10^{-3}$, while our numerical approach predicts $\varDelta ^{*}/c_n=\{7.59,10.2\} \times 10^{-3}$. Non-parallel flow effects are thus quite substantial and should not be neglected. We note, however, that receptivity amplitudes follow similar qualitative trends to those observed in the analysis of flat plate boundary layers undertaken by Choudhari & Streett (Reference Choudhari and Streett1992). Therefore we can instead choose $k$ such that the maximum of (3.13) occurs for the same value of $\varDelta$ as observed in the numerical results. The blue dash-dotted curve in figure 19(a) superimposed on the results for $s_b=0.30$ shows how this model fits the numerical results. Agreement is exceptional for small roughness widths, gradually deteriorating for longer roughness elements. This is to be expected since the longer the streamwise extent of the roughness due to which receptivity arises, the more likely it is that the parallel flow assumption is invalid.

Figure 19. Equivalent receptivity amplitude variation with the width of the Gaussian-shaped roughness element. Comparison for different positions of the roughness at three angles of attack ($\varTheta _i=45^{\circ }$, $\mathcal {S}=46.16$); (a) $\alpha =-2^{\circ }$, (b) $\alpha =0^{\circ }$ and (c) $\alpha =2^{\circ }$.

In figure 20 we show the equivalent results for three different frequencies at a fixed angle of attack $\alpha =0^{\circ }$. Receptivity amplitudes peak at smaller widths of the roughness element for increasing frequencies – this confirms the prediction of parallel flow theory in (3.14). This also explains the apparent decrease in receptivity amplitudes for a fixed-width roughness element shown in figure 18.

Figure 20. Equivalent receptivity amplitude variation with the width of the Gaussian-shaped roughness element. Comparison for different positions of the roughness at three frequencies ($\alpha =0^{\circ }$, $\varTheta _i=45^{\circ }$); (a) $\mathcal {S}=46.16$, (b) $\mathcal {S}=69.24$ and (c) $\mathcal {S}=115.4$.

3.5.3. Effects of sinusoidally distributed roughness wavelength

Acoustic receptivity in the presence of distributed roughness is a more interesting problem from a practical viewpoint because wing surface inhomogeneities can be characterised as a non-stationary random process, possessing irregularities over a wide range of wavelengths (Sayles & Thomas Reference Sayles and Thomas1978; Mughal & Ashworth Reference Mughal and Ashworth2013). The study of acoustic receptivity in the presence of localised roughness elements provides insight into the physical mechanisms of receptivity as well as a simpler mathematical problem. It can therefore be regarded as a cornerstone to understand the physics and study of the more general problem of continuous and discrete spectrum surface features. In fact, Choudhari & Streett (Reference Choudhari and Streett1992) proposed that receptivity to distributed roughness can be calculated as the sum of contributions from localised short-scale wall inhomogeneities. Alternatively, the study of surface features with a continuous spectrum can be decomposed into the study of sinusoidally distributed roughness via a Fourier series. This aspect motivates the results presented in this section.

The surface roughness geometry studied in this section is described in (3.3). Receptivity amplitudes as a function of the roughness wavelength are presented in figure 21 for various combinations of angle of attack and angle of incidence of the acoustic wave. The results are converged with respect to the length of the distributed roughness. This means that the roughness patch encompasses the whole region where the boundary layer is receptive and that the presence of roughness beyond this region would not alter the results.

Figure 21. Equivalent receptivity amplitude variation with the wavelength of sinusoidally distributed roughness. Comparison for different acoustic-wave angles of incidence at three angles of attack ($\mathcal {S}=46.16$). Boundary-layer instability wavelength at the first branch of neutral stability is marked with a vertical line; (a) $\alpha =-2^{\circ }$, (b) $\alpha =0^{\circ }$ and (c) $\alpha =2^{\circ }$.

Receptivity amplitudes are an order of magnitude larger for distributed roughness than for localised roughness, similarly to what has been found for flat plate geometries (Choudhari & Streett Reference Choudhari and Streett1992; Raposo et al. Reference Raposo, Mughal and Ashworth2018). One striking difference, however, is that for certain angles of incidence of the acoustic wave there is a very large band of wavelengths for which the boundary layer is receptive. One such example is for $\alpha =2^{\circ }$ and $\varTheta =45^{\circ }$, as opposed to the narrower bands observed for $\varTheta =180^{\circ }$ for example. Another difference is that maximal receptivity does not occur for wavelengths matching the boundary-layer instability wavelength. This is likely due to strong non-parallel flow effects.

Similar to the results in the previous sections, there continues to be no definitive trend when comparing receptivity at different angles of attack. Nonetheless, for most angles of incidence of the acoustic wave, amplitudes are higher at lower angles of attack. This indicates that boundary layers developing under more adverse pressure gradients are less receptive to T–S waves. This is in agreement with the findings of Crouch (Reference Crouch1994) and Raposo (Reference Raposo2020) for Falkner–Skan boundary layers. However, note that instabilities developing in adverse pressure gradients are known to be more amplified and therefore they might still be the leading cause of flow breakdown into turbulence, even though from a receptivity perspective favourable pressure gradients give rise to the largest boundary-layer disturbances.

Lastly, in figure 22 we consider the effects of varying frequency on receptivity amplitudes. It is shown that the boundary layer is undoubtedly more receptive to upstream-travelling acoustic waves than it is to its downstream-travelling counterparts. This observation is valid for all angles of attack and frequencies considered. Furthermore, an increase in frequency leads to higher receptivity amplitudes but also to markedly narrower bands of wavelengths to which the boundary layer is responsive. The fact that the boundary layer is both more receptive and more unstable for the frequency $\mathcal {S}=115.4$ is important from a transition prediction and control perspective. It supports the assumption that this frequency plays a key role in transition in this configuration but it also suggests that implementing control strategies to minimise receptivity or reduce instability growth may be effective in delaying the onset of turbulence.

Figure 22. Equivalent receptivity amplitude variation with the wavelength of sinusoidally distributed roughness. Comparison for different acoustic-wave angles of incidence at three frequencies ($\alpha =0^{\circ }$). Boundary-layer instability wavelength at the first branch of neutral stability is marked with a vertical line; (a) $\mathcal {S}=46.16$, (b) $\mathcal {S}=69.24$ and (c) $\mathcal {S}=115.4$.