1. Introduction
Our concern is about the convective instability that affects laminar boundary layers developing over streamwise-concave surfaces where the wall-normal pressure gradient of the disturbance cannot balance the centrifugal force. This inviscid mechanism leads to the formation of Görtler vortices, streamwise-oriented and spanwise-adjacent counterrotating structures that appear when the boundary-layer thickness is comparable to their spanwise wavelength. The rapid amplification of the velocity disturbances causes the breakdown to turbulence via secondary-instability mechanisms. In compressible boundary layers, temperature fluctuations also play a role in the transition process (Wang, Wang & Zhao Reference Wang, Wang and Zhao2018).
The accurate computation of the generation and evolution of Görtler vortices has posed theoretical challenges. A derivation of the equations governing the linear growth of Görtler vortices in concave-plate boundary layers was given by Floryan & Saric (Reference Floryan and Saric1979), who pointed out that the streamwise velocity component of the disturbances is asymptotically larger than the transverse components. Hall (Reference Hall1983) noted that the parabolic nature of these equations results in a critical dependence of the disturbance growth and neutral curves on the upstream flow conditions. In contrast to Tollmien–Schlichting (TS) waves of short streamwise wavelength, whose receptivity and growth occur in two distinct phases (Goldstein Reference Goldstein1983), the excitation and growth of streamwise-elongated Görtler vortices take place simultaneously (Wu, Zhao & Luo Reference Wu, Zhao and Luo2011). In general, the computation of the vortices requires the solution of an initial-boundary-value problem where the boundary-layer disturbances match the external perturbation flow (Wu et al. Reference Wu, Zhao and Luo2011; Xu, Ricco & Duan Reference Xu, Ricco and Duan2024). Amongst different external flow disturbances, free stream vortical disturbances have been recognised as powerful initiators of convective instability (Wu et al. Reference Wu, Zhao and Luo2011). Near the leading edge, the free stream turbulence generates laminar streaks that may evolve into Görtler vortices when the wall is concave in the streamwise direction. The effect of nonlinearity on incompressible Görtler vortices generated by free stream vortical disturbances was investigated by Xu, Zhang & Wu (Reference Xu, Zhang and Wu2017), Marensi & Ricco (Reference Marensi and Ricco2017) and Xu, Liu & Wu (Reference Xu, Liu and Wu2020) in the asymptotic framework of Leib, Wundrow & Goldstein (Reference Leib, Wundrow and Goldstein1999) (henceforth LWG99). Their accurate specification of the initial and outer boundary conditions resulted in good agreement with wind tunnel data. Compressible Görtler vortices engendered by free stream vortical disturbances were studied by Viaro & Ricco (Reference Viaro and Ricco2019a ) and Viaro & Ricco (Reference Viaro and Ricco2019b ). A comprehensive review of instability and transition involving compressible Görtler vortices is found in Xu et al. (Reference Xu, Ricco and Duan2024).
The persistence of laminar streaks and Görtler vortices in boundary layers and their role in laminar–turbulent transition calls for methodologies aimed at their attenuation. In the compressible regime, transition delay is desirable in boundary layers over high-speed vehicles and along supersonic wind-tunnel nozzles (Xu et al. Reference Xu, Ricco and Duan2024). In the first case, this delay leads to friction drag reduction and an attenuation of the thermal surface load. In the second case, the persistence of a laminar boundary layer over nozzles is a crucial step towards the design of quiet wind tunnel sections. We are therefore motivated to investigate wall suction as a viable control strategy for compressible boundary layers over streamwise-concave surfaces.
In the following subsections, we limit the scope of the literature review to wall suction for the attenuation of laminar streaks and Görtler vortices in laminar boundary layers.
1.1. Experimental studies
The use of transpiration devices on wind-tunnel walls dates back to the 1960s (Schneider Reference Schneider2008a
). Efforts to delay transition in supersonic wind tunnels were undertaken using longitudinal slots (Beckwith & Bertram Reference Beckwith and Bertram1972; Beckwith et al. Reference Beckwith, Harvey, Harris and Holley1973) and perforated plates (Leontiev & Pavlyuchenko Reference Leontiev and Pavlyuchenko2008). An issue with transpiration surfaces is related to their non-uniformities and defects. These imperfections are known to generate isolated or distributed roughness effects to which the thinner suction boundary layer may be susceptible (Saric Reference Saric1985; Messing & Kloker Reference Messing and Kloker2010). The impact of distributed roughness on high-speed transition is, however, still poorly understood (Schneider Reference Schneider2008b
; Running et al. Reference Running, Bemis, Hill, Borg, Redmond, Jantze and Scalo2023). While relatively large roughness (
$50$
–
$500\,{\unicode{x03BC}} {\textrm {m}}$
) dramatically triggers the breakdown to turbulence in the absence of suction (Bountin et al. Reference Bountin, Gromyko, Maslov, Polivanov and Sidorenko2016; Gui et al. Reference Gui, Zhang, Li, Xu and Wu2023), Ludwieg-tube measurements at subsonic speeds (Dimond et al. Reference Dimond, Costantini, Risius, Klein and Rein2020, Reference Dimond, Costantini and Klein2022) and experiments in incompressible wind tunnels (Methel et al. Reference Methel, Forte, Vermeersch and Casalis2021) suggest that suction effectively delays transition even in the presence of suction holes as large as
$100\,{\unicode{x03BC}} {\textrm {m}}$
.
Fransson & Alfredsson (Reference Fransson and Alfredsson2003) applied uniform suction through a sintered-plastic plate with small pores (
${\cong}10\,{\rm \unicode{x03BC}} {\textrm {m}}$
) and reported an attenuation of TS waves and laminar streaks at low speeds and high free stream turbulence levels. Their results show a gradual evolution of the Blasius boundary layer to the asymptotic-suction boundary layer (ASBL) over the porous plate. Despite a twofold reduction in the boundary-layer thickness, the spanwise wavelength of the streaks
$\lambda _z^\ast$
was not affected. Yoshioka, Fransson & Alfredsson (Reference Yoshioka, Fransson and Alfredsson2004) measured
$\lambda _z^\ast$
in a series of experiments with suction, concluding that its value was primarily determined by the shape of the leading edge and the oncoming free stream perturbation flow. Kurian & Fransson (Reference Kurian and Fransson2011) investigated the evolution of disturbances induced in an ASBL by an array of roughness elements. Their findings showed good agreement with optimal growth predictions when the optimisation time was chosen to fit the experimental data.
The experimental studies on the influence of suction on Görtler vortices are very limited. Mangalam, Dagenhart & Kalburgi (Reference Mangalam, Dagenhart and Kalburgi1987) placed spanwise-aligned rods on the concave section of a wing profile and reported an attenuation of the velocity fluctuations associated with Görtler vortices. A selective transpiration technique was employed by Myose & Blackwelder (Reference Myose and Blackwelder1995) on a concave boundary layer. They applied suction at periodic spanwise locations where the low-speed streaks were known to develop and reported a delay in the breakdown of the Görtler vortices.
To the best of the authors’ knowledge, there are no experimental investigations on compressible Görtler vortices evolving in suction boundary layers. Li et al. (Reference Li, Choudhari, Paredes, Schneider and Portoni2018) examined the use of uniform suction to mitigate Görtler instability on flared cones at zero angle of attack in a hypersonic wind tunnel. To that end, they manufactured a model comprising an aft cone and a rear flared cone with a truncated permeable volume installed on the aft section, although no measurements of the perturbed flow in the boundary layer were conducted. Furthermore, wall suction has never been used to attenuate Görtler vortices in boundary layers along the nozzles of supersonic wind tunnels, over the external wetted surfaces of aircraft or over the internal surfaces of aircraft engine intakes. Suction actuators have successfully delayed transition in boundary layers on aircraft wings, yet with no evidence that the perturbations involved were Görtler vortices (Krishnan, Bertram & Seibel Reference Krishnan, Bertram and Seibel2017).
1.2. Studies based on direct numerical simulations
Sescu, Alaziz & Afsar (Reference Sescu, Alaziz and Afsar2018) and Sescu, Alaziz & Afsar (Reference Sescu, Alaziz and Afsar2019) investigated the effect of localised blowing and suction by performing direct numerical simulations (DNSs) of a supersonic concave boundary layer. They implemented a feedback control algorithm to attenuate the nonlinear growth of the vortices. Direct numerical simulations were also employed to study how wall transpiration modifies perturbed Mach-6 boundary layers over flared cones, by Hader & Fasel (Reference Hader and Fasel2021) for the case of alternate blowing and suction and by Hollender, Dwivedi & Candler (Reference Hollender, Dwivedi and Candler2019) for the case of distributed suction, with the objective to reproduce the flow configuration studied experimentally by Li et al. (Reference Li, Choudhari, Paredes, Schneider and Portoni2018).
1.3. Theoretical studies
The effect of uniform wall transpiration on the linear stability of Görtler vortices has been studied for self-similar and asymptotic suction (Kobayashi Reference Kobayashi1972, Reference Kobayashi1974; El-Hady & Verma Reference El-Hady and Verma1981, Reference El-Hady and Verma1984; Floryan & Saric Reference Floryan and Saric1983). Self-similar suction, which features an unbounded wall-normal velocity as the leading edge is approached, has never been implemented in the laboratory. Self-similar solutions offer the advantage of being more tractable theoretically and often provide valuable insights into stability and transition, especially when solving a system of partial differential equations is not feasible (Stewartson Reference Stewartson1964; Anderson Reference Anderson2019; Al-Malki et al. Reference Al-Malki, Hussain, Garrett and Calabretto2021). Ricco & Dilib (Reference Ricco and Dilib2010) and Ricco, Shah & Hicks (Reference Ricco, Shah and Hicks2013) imposed self-similar suction on incompressible and compressible laminar base flows to study the receptivity and growth of streaks and oblique TS waves within the asymptotic framework of Ricco & Wu (Reference Ricco and Wu2007) and LWG99. Marensi & Ricco (Reference Marensi and Ricco2017) solved the nonlinear boundary-region equations in the asymptotic framework of LWG99 and reported a marked attenuation of the Görtler vortices in the presence of spanwise-dependent wall suction. Es-Sahli et al. (2023) applied an optimal-control technique to the nonlinear boundary-region equations and reported a marked attenuation of the energy growth. Within these theoretical frameworks, uniform wall suction has never been utilised to attenuate the growth of Görtler vortices at any Mach number.
The laminar flow modified by uniform suction is initially streamwise-dependent, the asymptotic-suction solution being valid only sufficiently downstream of the suction onset. This solution was derived for the first time by Griffith & Meredith (Reference Griffith and Meredith1936) and extended to the compressible regime by Young (Reference Young1948). Morduchow (Reference Morduchow1963) showed that the ASBL solution remains valid in the presence of streamwise pressure gradients. An exact compressible solution was derived by Lew & Fanucci (Reference Lew and Fanucci1955) for the case where the wall suction was uniform at any streamwise location, while Görtler (Reference Görtler1957) treated the case of arbitrary distributions of the suction velocity in the incompressible regime.
1.4. Outline of the study
We aim to study the growth of compressible Görtler vortices developing on a concave wall with steady uniform suction applied downstream of an impermeable region. The receptivity of the base flow to external perturbations is a central ingredient of our framework, i.e. the Görtler vortices are generated by – and continuously exposed to – free stream vortical disturbances. Our analysis is concerned with the flow configurations and disturbance wavelengths commonly reported in the experimental literature (e.g. de Luca et al. Reference de Luca, Cardone, Aymer de la Chevalerie and Fonteneau1993; Ciolkosz & Spina Reference Ciolkosz and Spina2006; Li et al. Reference Li, Choudhari, Paredes, Schneider and Portoni2018; Wang et al. Reference Wang, Wang and Zhao2018). The theoretical framework of LWG99, adapted by Wu et al. (Reference Wu, Zhao and Luo2011) and Viaro & Ricco (Reference Viaro and Ricco2019a ) to the case of self-similar boundary layers over concave walls, is thus extended to and solved for non-similar compressible boundary layers for the first time.
We shall consider the case where uniform suction is applied downstream of an impermeable region. The velocity and temperature distributions of the laminar base flow do not obey self-similarity and the evolution of the laminar flow must be described by the non-similar boundary-layer equations (e.g. Cebeci Reference Cebeci2002), as shown by Fransson & Alfredsson (Reference Fransson and Alfredsson2003) in the incompressible case. Furthermore, the receptivity of the base flow to free stream vortical disturbances plays a critical role in the vicinity of the leading edge where the suction velocity is zero or finite. This effect is another compelling reason why neither the self-similar suction model nor the ASBL model can provide an accurate description of the base flow of interest herein.
The asymptotic framework, the governing equations of the base flow and the disturbance flow, and the boundary conditions are discussed in § 2. The validation of the non-similar laminar base flow, the effect of suction on the growth of the compressible Görtler vortices and the behaviour of the neutral stability curves are discussed in § 3. Conclusions are drawn in § 4.
Schematic of the flow system. Görtler vortices grow over a concave plate under the continuous forcing of free stream vorticity. The plate is infinitely thin at the leading edge and uniform suction is applied in the yellow region
$\hat {x}\gt \hat {x}_s$
. The disturbance flow is periodic along the spanwise (
$z^\ast$
) direction and the plate has an infinite spanwise width.
2. Mathematical formulation
As shown in figure 1, we consider a uniform flow of velocity
$U_\infty ^\ast$
and temperature
$T_\infty ^\ast$
past a thin concave plate with radius of curvature
$r_c^\ast$
. This flow generates a thin boundary layer of thickness
$\delta ^\ast \ll r_c^\ast$
over the plate. Wall suction is applied by imposing a streamwise-dependent wall-normal velocity
$V_w$
at the wall. Downstream of an impermeable region (grey area in figure 1) followed by a thin adjustment region centred at
$x_s^\ast$
(red area in figure 1), the wall suction velocity is uniform (yellow area in figure 1). The flow is described in a curvilinear, orthogonal system of coordinates
$\left \lbrace x^\ast ,y^\ast ,z^\ast \right \rbrace$
centred at the leading edge. The streamwise coordinate
$x^\ast$
follows the curvature of the plate, and
$y^\ast$
and
$z^\ast$
denote the wall-normal and spanwise coordinates, respectively. The flow is spanwise-periodic and the concave plate has an infinite spanwise width. All dimensional quantities are denoted by the superscript
$\ast$
. The governing equations are derived by introducing the Lamé coefficients
$\lbrace 1-y^\ast /r_c^\ast ,1,1\rbrace$
in the Navier–Stokes equations (e.g. El-Hady & Verma Reference El-Hady and Verma1984; Viaro & Ricco Reference Viaro and Ricco2019a
). The spatial lengths are normalised using
$\varLambda ^\ast = \lambda _z^\ast / 2\pi$
, where
$\lambda _z^\ast$
denotes the spanwise wavelength of the disturbances (LWG99). Unlike the boundary-layer thickness,
$\lambda _z^\ast$
remains constant in both the free stream and in the boundary layer, regardless of whether suction is applied or not (Tani Reference Tani1962; Finnis & Brown Reference Finnis and Brown1997; Fransson & Alfredsson Reference Fransson and Alfredsson2003; Yoshioka et al. Reference Yoshioka, Fransson and Alfredsson2004; Tandiono, Winoto & Shah Reference Tandiono, Winoto and Shah2008). Furthermore, laminar streaks and Görtler vortices emerge at downstream locations where the boundary-layer thickness has grown to a magnitude comparable to
$\lambda _z^\ast$
, making
$\lambda _z^\ast$
the natural length scale of the problem (Swearingen & Blackwelder Reference Swearingen and Blackwelder1987; Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Wu et al. Reference Wu, Zhao and Luo2011). The velocity components
$\left \lbrace u^\ast ,v^\ast ,w^\ast \right \rbrace$
, the density
$\rho ^\ast$
and the temperature
$T^\ast$
scale with their reference free stream values,
$U_\infty ^\ast$
,
$\rho _\infty ^\ast$
and
$T_\infty ^\ast$
. The pressure is normalised by
$\rho _\infty ^\ast U_\infty ^{\ast 2}$
and the time is scaled by
$\varLambda ^\ast /U_\infty ^\ast$
. The Reynolds number is
${\textit{Re}} = \rho _\infty ^\ast U_\infty ^\ast \varLambda ^\ast /{\mu} _\infty ^\ast \gg 1$
, while the Mach number
${\textit{Ma}} = U_\infty ^\ast / (\gamma {\mathcal{R}}^\ast T_\infty ^\ast )^{1/2}=O(1)$
, where
${\mu}^\ast$
is the dynamic viscosity,
$\gamma =1.4$
is the heat capacity ratio and
${\mathcal{R}}^\ast =287.05\,{\textrm {J}}\,{\textrm {kg}}^{-1}\,{\textrm {K}}^{-1}$
is the perfect gas constant of air. The Prandtl number is
$Pr = \gamma (\gamma -1 )^{-1} {\mu}_\infty ^\ast {\mathcal{R}}^\ast /k_\infty ^\ast =0.71$
, where
$k^\ast$
is the thermal conductivity. Scaled quantities are not indicated by any symbol.
Schematic of the four asymptotic regions introduced by LWG99 and their domains (dashed lines). The disturbance flow is inviscid in region I (orange) and viscous in regions II, III and IV (blue). The boundary layer thickness is shown in grey. The laminar streaks and the Görtler vortices develop in region III, where the flow is governed by the compressible linearised unsteady boundary-region equations (CLUBR) introduced in § 2.2. The solution to the far-field (
$\eta \gg 1$
) form of the CLUBR equations matches the inviscid flow in region I (2.1) for
$\hat {x}\to 0$
and the inner limit of region IV (
$\hat {y}\to 0$
) for large
$y\to \infty$
. The size of the regions is not to scale.
We follow the receptivity framework first introduced in LWG99. The framework relies on the assumption of large-
${\textit{Re}}$
flows and streamwise-elongated, spanwise-periodic disturbances with a small streamwise wavenumber
$k_x\ll 1$
. It is convenient to introduce the scaled variables
$\hat {x}=x/{\textit{Re}}$
and
$\hat {t}=t/{\textit{Re}}$
. A distinguished scaling emerges and the domain splits in the four asymptotic regions shown in figure 2. Regions I and II are near the leading edge, at
$\hat {x}\ll 1$
(but
$1\ll x\ll {\textit{Re}}$
), where the boundary-layer thickness is smaller than the spanwise wavelength of the disturbance, i.e.
$\delta = \delta ^\ast /\varLambda ^\ast \ll 1$
. The viscous effects are confined in this thin boundary layer in region II, where the disturbance flow is governed by the unsteady boundary-layer equations. Small-amplitude, non-interacting vortical disturbances are passively advected by the inviscid flow in region I, where
$y=O(1)$
. They take the form of monochromatic perturbations of the gust type,
\begin{equation} \boldsymbol{u} - \boldsymbol{i} = \epsilon \boldsymbol{\hat {u}}^\infty {\textrm{e}}^{{\textrm{i}}\left (\boldsymbol{k} \boldsymbol{\cdot }\boldsymbol{x} - {\mathcal{F}}\hat {t}\right )} + \textrm{c.c.}, \end{equation}
where
$\epsilon \ll 1$
,
$\boldsymbol{\hat {u}}^\infty =\lbrace \hat u^\infty , v^\infty , \hat w^\infty \rbrace$
is the gust amplitude,
$\boldsymbol{k}=\lbrace k_x,k_y,k_z\rbrace$
is the wavenumber vector,
$\boldsymbol{i}$
is the streamwise (
$x^\ast$
) vector and c.c. indicates the complex conjugate. The parameter
${\mathcal{F}} = k_x{\textit{Re}} = \rho _\infty ^\ast 2\pi f^\ast \varLambda ^{\ast 2}/{\mu}_\infty ^\ast = O(1)$
is the scaled disturbance frequency
$f^\ast$
.
Regions III and IV are located farther downstream at
$\hat {x}=O(1)$
. There,
$\delta ^\ast /\varLambda ^\ast = O(1)$
, the spanwise and wall-normal viscous diffusion are comparable and low-frequency, streamwise-elongated gusts induce three-dimensional disturbances within the boundary layer. Viscous effects are confined within the thicker boundary layer in region III, where
$y=O(1)$
and the laminar streaks and the Görtler vortices ensue. The evolution of these structures is governed by the unsteady boundary-region equations, which represent the rigorous asymptotic limit of the Navier–Stokes equations for
$k_x \ll 1$
and
${\textit{Re}} \gg 1$
. The large disparity between spanwise and streamwise scales results in
$O(\epsilon )$
free stream fluctuations generating
$O(\epsilon {\textit{Re}})$
streamwise velocity disturbances in region III. The inviscid outer flow is influenced by the boundary-layer displacement in region IV, where
$y = O({\textit{Re}})$
and the gusts (2.1) decay as they are advected downstream.
For asymptotically small disturbance amplitude
$r_t=\epsilon {\textit{Re}}\ll 1$
, the unsteady, three-dimensional disturbance flow
$\boldsymbol{\grave {q}}$
is linearised about the steady, two-dimensional laminar base flow
$\boldsymbol{Q}$
, i.e.
$\boldsymbol{q} (\hat {x},y,z,\hat {t} ) = \boldsymbol{Q} (\hat {x},y )+\epsilon \boldsymbol{\grave {q}} (\hat {x},y,z,\hat {t} )$
:
\begin{equation} \!\! \lbrace u,v,w,\tau ,p\rbrace (\hat {x},y,z,\hat {t}\kern1pt) = \left \lbrace U,V,0,T,\frac {1}{\gamma {\textit{Ma}}^2}\right \rbrace (\hat {x},y) + \epsilon \lbrace \grave u, \grave v, \grave w, \grave \tau , \grave p \rbrace (\hat {x},y,z,\hat {t}\kern1pt). \end{equation}
In our case of uniform suction downstream of an impermeable region, the base flow
$\boldsymbol{Q}$
is not self-similar. It is governed by a system of parabolic partial differential equations, the compressible Blasius solution being valid only along the impermeable region
$\hat {x}\lt \hat {x}_s$
.
2.1. The non-similar laminar base flow
We now obtain the non-similar, compressible boundary-layer equations with emphasis on the derivation of the wall-normal velocity component
$V$
. To the best of the authors’ knowledge, this derivation is not available in the literature. We first define a stream function
$\psi (\hat {x},y )$
that satisfies the continuity equation
$\rho U = \partial \psi /\partial y$
and
$\rho V = - {\textit{Re}}^{-1}\partial \psi /\partial \hat {x}$
. The governing equations are derived in terms of the normalised stream function
$F (\hat {x},\eta ) = \psi (\hat {x},y ) (2\hat {x} )^{-1/2}$
and the similarity variable
$\eta (\hat {x},y) = \bar Y(\hat {x},y) (2\hat {x} )^{-1/2}$
, where
\begin{equation} \bar Y (\hat {x},y) \equiv \int _0^y [T(\hat {x},\breve y)]^{-1}{\textrm{d}}\breve y \end{equation}
is the Dorodnitsyn–Howarth variable (Stewartson Reference Stewartson1964). Inverting (2.3) yields
\begin{equation} y(\hat {x},\eta) = (2\hat {x})^{1/2}\int _0^{\eta } T(\hat {x},\breve \eta) {\textrm{d}}\breve \eta . \end{equation}
The use of
$\eta$
is convenient because the boundary layer is self-similar along the impermeable region and because the initial and outer boundary conditions are more readily specified in terms of
$\eta$
(Xu et al. Reference Xu, Liu and Wu2020). The streamwise momentum and static-enthalpy balance equations that govern the base flow
$\boldsymbol{Q}$
are (e.g. Cebeci Reference Cebeci2002)
\begin{align} \left (\frac {{\mu} }{T}F^{\prime \prime }\right )^\prime + FF^{\prime \prime } &= 2\hat {x}\left (F^\prime \left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _{\eta } - F^{\prime \prime }\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\!, \end{align}
\begin{align} FT^\prime + (\gamma -1){\textit{Ma}}^2 \frac {{\mu} }{T}(F^{\prime \prime })^2 + \frac {1}{\textit{Pr}}\left (\frac {{\mu} }{T}T^\prime \right )^\prime &= 2\hat {x}\left (F^\prime \left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta - T^\prime \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\!, \end{align}
where the prime denotes differentiation in
$\eta$
and the viscosity
${\mu}$
is modelled with Sutherland’s law. The wall-normal velocity component of the base flow
$V$
(2.2) is found by applying the chain rule to
$\partial \psi /\partial \hat {x}$
and introducing the perfect gas equation
$\rho T=1$
,
\begin{equation} V(\hat {x},\eta) = -\frac {T}{{\textit{Re}}}\frac {\partial \psi }{\partial \hat {x}} = -\frac {1}{{\textit{Re}}(2\hat {x})^{1/2}}\left (TF + 2\hat {x} T\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta + 2 \hat {x} T\left .\frac {\partial \eta }{\partial \hat {x}}\right \vert _yF^\prime \right )\!. \end{equation}
The unknown term
$\left .\partial \eta /\partial \hat {x}\right \vert _y$
is found by recalling that
$\hat {x}$
and
$y$
are orthogonal, and thus the total derivative of (2.4) with respect to
$\hat {x}$
is null. Using the chain rule yields
\begin{equation} \frac {\partial y(\hat {x},\eta)}{\partial \hat {x}} = \left .\frac {\partial y(\hat {x},\eta)}{\partial \hat {x}}\right \vert _\eta + \left .\frac {\partial \eta (\hat {x},y)}{\partial \hat {x}}\right \vert _y \left .\frac {\partial y (\hat {x},\eta)}{\partial \eta }\right \vert _{\hat {x}} = 0, \end{equation}
where
\begin{align} \left .\frac {\partial y(\hat {x},\eta)}{\partial \hat {x}}\right \vert _\eta &= \frac {1}{(2\hat {x})^{1/2}}\int _0^\eta T(\hat {x},\breve \eta) {\textrm{d}}\breve \eta + (2\hat {x})^{1/2}\int _0^\eta \left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta {\textrm{d}}\breve \eta , \end{align}
\begin{align} \left .\frac {\partial y (\hat {x},\eta)}{\partial \eta }\right \vert _{\hat {x}} &= (2\hat {x})^{1/2} T (\hat {x},\eta). \end{align}
The derivative (2.8a
) represents the rate of change in
$y$
due to a change in
$\hat {x}$
while moving along a path at fixed
$\eta$
. The derivative (2.8b
) is the rate of change in
$y$
due to a variation in
$\eta$
while
$\hat {x}$
is fixed. The sought derivative
$\left .\partial \eta /\partial \hat {x}\right \vert _y = - \left .\partial y/\partial \hat {x}\right \vert _\eta (\left .\partial y/\partial \eta \right \vert _{\hat {x}} )^{-1}$
is
\begin{equation} \left .\frac {\partial \eta (\hat {x},y)}{\partial \hat {x}}\right \vert _y = - \frac {1}{2\hat {x} T (\hat {x},\eta)}\int _0^\eta T (\hat {x},\breve \eta) {\textrm{d}}\breve \eta - \frac {1}{T (\hat {x},\eta)}\int _0^{\eta }\left .\frac {\partial T}{\partial \hat {x}}\right \vert _{\breve \eta }{\textrm{d}}\breve \eta \equiv - \frac {\eta _c}{2 \hat {x}}, \end{equation}
where the compressible similarity variable
$\eta _c (\hat {x},\eta )$
,
\begin{equation} \eta _c (\hat {x},\eta) \equiv \frac {1}{T (\hat {x},\eta)}\int _0^\eta \left [T(\hat {x},\breve {\eta }) + 2\hat {x}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _{\breve \eta }\right ]{\textrm{d}}\breve {\eta }, \end{equation}
is defined. Equation (2.10) reduces to
$\eta _c(\eta ) = T^{-1}\int _0^{\eta }T{\textrm{d}}\breve \eta$
in the self-similar case (Viaro & Ricco Reference Viaro and Ricco2019a
). The self-similar form of (2.10) can also be derived by expanding the integral in (3.1.5) in Stewartson (Reference Stewartson1964, p. 36).
The base-flow velocity components
\begin{align} U (\hat {x},\eta) &= F^\prime , \end{align}
\begin{align} V (\hat {x},\eta) &=\frac {T}{{\textit{Re}}(2\hat {x})^{1/2}}\left (\eta _cF^\prime - F - 2\hat {x} \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\!, \end{align}
reduce to (2.7) in Viaro & Ricco (Reference Viaro and Ricco2019a
) in the self-similar case. The wall is adiabatic,
$T^\prime (\hat {x},0 ) = 0$
. A no-slip condition is imposed on the streamwise velocity component,
$F^\prime (\hat {x},0 ) = 0$
, while the effect of the transpiration velocity
$V_w=V (\hat {x},0 )$
is described by the equation
\begin{equation} \left [F + 2\hat {x} \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta + \frac {\gamma _w (2\hat {x})^{1/2}}{T} \right ](\hat {x},0) = 0, \end{equation}
where
$\gamma _w (\hat {x} ) = V_w{\textit{Re}} = O(1)$
is the suction parameter. We use the incompressible definition of
$\gamma _w$
(Floryan & Saric Reference Floryan and Saric1983) rather than the compressible one (El-Hady & Verma Reference El-Hady and Verma1984) because
$T (\hat {x},0 )$
is not constant. The non-dimensional suction velocity
$V_w\ll 1$
is also known as the suction coefficient in the experimental literature (e.g. Myose & Blackwelder Reference Myose and Blackwelder1995; Fransson & Alfredsson Reference Fransson and Alfredsson2003; Leontiev & Pavlyuchenko Reference Leontiev and Pavlyuchenko2008) and typically ranges between
$10^{-5}$
and
$10^{-3}$
. The velocity and the temperature are uniform in the external region, i.e.
$F^\prime \to 1$
and
$T\to 1$
for
$\eta \gg 1$
.
In the limit of large streamwise distance, the two-dimensional steady boundary-layer equations tend to the compressible asymptotic-suction (ASBL) system of ordinary differential equations (Young Reference Young1948; Morduchow Reference Morduchow1963),
\begin{align} \frac {{\textrm{d}}}{{\textrm{d}} N}\left (\frac {{\mu} }{T}\frac {{\textrm{d}} U}{{\textrm{d}} N} + U\right ) &= 0, \end{align}
\begin{align} \frac {1}{\textit{Pr}}\frac {{\textrm{d}}}{{\textrm{d}} N}\left (\frac {k}{T}\frac {{\textrm{d}} T}{{\textrm{d}} N} + T\right ) + \left (\gamma -1\right ){\textit{Ma}}^2\frac {{\mu} }{T}\left (\frac {{\textrm{d}} U}{{\textrm{d}} N}\right )^2 &= 0, \end{align}
where
$N=-\gamma _w\bar Y/T_w$
,
$\bar Y$
is defined in (2.3) and
$T_w$
is the wall temperature.
2.2. The unsteady, three-dimensional disturbance flow
The disturbance flow
$\boldsymbol{\grave {q}}$
in region III is governed by the CLUBR. By substituting the expansion (2.2) with the disturbances (Floryan & Saric Reference Floryan and Saric1979; Leib et al. Reference Leib, Wundrow and Goldstein1999; Wu et al. Reference Wu, Zhao and Luo2011)
\begin{equation} \boldsymbol{\grave {q}} = {\textrm{i}} k_z\check {w}{\textit{Re}} \left \{\bar u, \frac {(2\hat {x})^{1/2}}{{\textit{Re}}} \bar v,\frac {\bar w}{{\textrm{i}} k_z{\textit{Re}}},\bar \tau ,\frac {\bar p}{{\textit{Re}}^2} \right \} {\textrm{e}}^{{\textrm{i}}\left (k_zz - {\mathcal{F}}\hat {t}\right )} \end{equation}
– where
$\check w = \hat w^\infty + {\textrm{i}} k_z \hat v^\infty / (k_x^2+k_z^2 )^{1/2} = O(1)$
– into the compressible Navier–Stokes equations, and by assuming
$r_c=r_c^\ast /\varLambda ^\ast =O({\textit{Re}}^2)$
, the CLUBR continuity
${\mathcal{C}}$
, streamwise momentum
${\mathcal{X}}$
, wall-normal momentum
$\mathcal{Y}$
, spanwise momentum
${\mathcal{Z}}$
and enthalpy
${\mathcal{E}}$
balances are recovered,
\begin{align} \text{$\left .{\mathcal{C}}\right \vert $ } & \left ( \frac {\eta _cT^\prime }{2\hat {x} T} - \frac {1}{T}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta \right ) \bar {u} + \left .\frac {\partial \bar {u}}{\partial \hat {x}}\right \vert _\eta - \frac {\eta _c}{2\hat {x}} \frac {\partial \bar {u}}{\partial \eta } - \frac {T^\prime }{T^2}\bar {v} + \frac {1}{T}\frac {\partial \bar {v}}{\partial \eta } + \bar {w} + \left (\frac {{\textrm{i}}{\mathcal{F}}}{T}\right . \nonumber \\& \left .+ \frac {F^\prime }{T^2}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta - \frac {T^\prime }{T^2}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta - \frac {FT^\prime }{2\hat {x} T^2}\right )\bar {\tau } - \frac {F^\prime }{T}\left .\frac {\partial \bar {\tau }}{\partial \hat {x}}\right \vert _\eta + \left (\frac {F}{2\hat {x} T} + \frac {1}{T}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\frac {\partial \bar {\tau }}{\partial \eta } = 0, \end{align}
\begin{align} \text{$\left .{\mathcal{X}}\right \vert $ } &\left (k_z^2{\mu} T - {\textrm{i}} {\mathcal{F}} - \frac {\eta _cF^{\prime \prime }}{2\hat {x}} + \left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _\eta \right )\bar u + F^\prime \left .\frac {\partial \bar u}{\partial \hat {x}}\right \vert _\eta - \left (\frac {F}{2\hat {x}} + \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\frac {\partial \bar u}{\partial \eta } \nonumber \\ & + \frac {1}{2\hat {x}}\left (\frac {{\mu} }{T}\frac {\partial \bar u}{\partial \eta }\right )^\prime + \frac {F^{\prime \prime }}{T} \bar v + \left (\frac {FF^{\prime \prime }}{2\hat {x} T} - \frac {F^\prime }{T}\left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _\eta + \frac {F^{\prime \prime }}{T}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\bar \tau - \left (\frac {{\mu} ^\prime F^{\prime \prime }}{2\hat {x} T}\bar \tau \right )^\prime = 0, \end{align}
\begin{align} \text{$\left .{\mathcal{Y}}\right \vert $ } & \left (\frac {TF-\eta _cTF^\prime +\eta _cT^\prime F -\eta _c^2TF^{\prime \prime }}{4\hat {x}^2} - \left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta - T\left .\frac {\partial ^2F}{\partial \hat {x}^2}\right \vert _\eta - \frac {F}{2\hat {x}}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta \right .\nonumber \\ & \left .+ \frac {\eta _cT^\prime - 2T}{2\hat {x}}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta + \frac {TF^\prime }{2\hat {x}}\left .\frac {\partial \eta _c}{\partial \hat {x}}\right \vert _\eta + \frac {\eta _cT}{\hat {x}}\left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _\eta + \frac {2 {G} F^\prime }{(2\hat {x})^{1/2}} \right ) \bar u + \frac {{\mu} ^\prime T^\prime }{3\hat {x}}\left .\frac {\partial \bar u}{\partial \hat {x}}\right \vert _\eta \nonumber \\ & + \!\left [\frac {\left (\eta _c{\mu} \right )^\prime }{12\hat {x}^2} - \frac {{\mu} ^\prime }{2\hat {x}}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta \right ]\frac {\partial \bar u}{\partial \eta } - \frac {{\mu} }{6\hat {x}}\frac {\partial }{\partial \hat {x}}\left (\frac {\partial \bar u}{\partial \eta }\right )_\eta + \frac {\eta _c{\mu} }{12\hat {x}^2}\frac {\partial ^2\bar u}{\partial \eta ^2} + \left (k_z^2{\mu} T \!-\! {\textrm{i}} {\mathcal{F}} \!+\! \frac {F^\prime }{2\hat {x}} \!-\! \frac {FT^\prime }{2\hat {x} T} \right .\nonumber \\ & \left .- \frac {T^\prime }{T}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta - \left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _\eta + \frac {F^\prime }{T}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta + \frac {\eta _cF^{\prime \prime }}{2\hat {x}} \right ) \bar v + F^\prime \left .\frac {\partial \bar v}{\partial \hat {x}}\right \vert _\eta - \left (\frac {F}{2\hat {x}} + \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\frac {\partial \bar v}{\partial \eta } \nonumber \\ & - \left (\frac {2{\mu} }{3\hat {x} T}\frac {\partial \bar v}{\partial \eta }\right )^\prime + \frac {{\mu} ^\prime T^\prime }{3\hat {x}}\bar w - \frac {{\mu} }{6\hat {x}} \frac {\partial \bar w}{\partial \eta } + \left \lbrace \frac {\eta _c(FF^\prime)^\prime }{4\hat {x}^2} - \frac {FF^\prime }{4\hat {x}^2} - \frac {F^2T^\prime }{4\hat {x}^2T} - \frac {{G}}{(2\hat {x})^{1/2}}\frac {(F^\prime)^2}{T} \right .\nonumber \\ & + \left (\frac {{\mu} ^\prime T^\prime F}{3\hat {x}^2 T}\right )^\prime - \frac {(\eta _c{\mu} ^\prime F^{\prime \prime})^\prime }{4\hat {x}^2} + \left [\frac {F^\prime T - FT^\prime }{2\hat {x} T} + \frac {\eta _cF^{\prime \prime }}{2\hat {x}} + \left (\frac {2{\mu} ^\prime T^\prime }{3\hat {x} T}\right )^\prime \right ]\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta + \left (\frac {2{\mu} ^\prime T^\prime }{3\hat {x} T} \right . \nonumber \\ & \left .+ \frac {{\mu} ^{\prime \prime }T^\prime }{\hat {x}} - \frac {F}{2\hat {x}} - \frac {\eta _cF^\prime }{2\hat {x}} \right )\left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _\eta + \frac {{\mu} ^\prime }{2\hat {x}}\left .\frac {\partial F^{\prime \prime }}{\partial \hat {x}}\right \vert _\eta + \left [\frac {FF^\prime }{\hat {x} T} - \frac {\eta _c(F^\prime)^2}{2\hat {x} T} - \left (\frac {2{\mu} ^\prime F^\prime }{3\hat {x} T}\right )^\prime \right . \nonumber \\ & \left . - \frac {{\mu} ^{\prime \prime }F^{\prime \prime }}{2\hat {x}} \right ]\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta + \frac {2F^\prime }{T}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta + F^\prime \left .\frac {\partial ^2F}{\partial \hat {x}^2}\right \vert _\eta - \frac {(F^\prime)^2}{2\hat {x}}\left .\frac {\partial \eta _c}{\partial \hat {x}}\right \vert _\eta - \frac {T^\prime }{T}\left (\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )^2 \nonumber \\ & \left .- \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _\eta - \frac {2{\mu} ^\prime F^\prime }{3\hat {x} T}\left .\frac {\partial T^\prime }{\partial \hat {x}}\right \vert _\eta \right \rbrace \bar \tau -\frac {{\mu} ^\prime F^{\prime \prime }}{2\hat {x}} \left .\frac {\partial \bar \tau }{\partial \hat {x}}\right \vert _\eta + \left (\frac {{\mu} ^\prime T^\prime F}{3\hat {x}^2T} - \frac {\eta _c{\mu} ^\prime F^{\prime \prime }}{4\hat {x}^2} \right .\nonumber \\ & \left .+ \frac {2{\mu} ^\prime T^\prime }{3\hat {x}}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta + \frac {{\mu} ^\prime }{\hat {x}}\left .\frac {\partial F^\prime }{\partial \hat {x}}\right \vert _\eta - \frac {2{\mu} ^\prime F^\prime }{3\hat {x} T}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta \right ) \frac {\partial \bar \tau }{\partial \eta } + \frac {1}{2\hat {x}}\frac {\partial \bar p}{\partial \eta } = 0, \end{align}
\begin{align} \text{$\left .{\mathcal{Z}}\right \vert $ } & \left ( k_z^2T{\mu} ^\prime \left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta - k_z^2\frac {\eta _c{\mu} ^\prime TT^\prime }{2\hat {x}}\right ) \bar u + \frac {k_z^2{\mu} T}{3} \left .\frac {\partial \bar u}{\partial \hat {x}}\right \vert _\eta -k_z^2 \frac {\eta _c{\mu} T}{6\hat {x}} \frac {\partial \bar u}{\partial \eta } + k_z^2{\mu} ^\prime T^\prime \bar v \nonumber \\ & + \frac {k_z^2{\mu} }{3} \frac {\partial \bar v}{\partial \eta } - \left ({\textrm{i}} {\mathcal{F}} - \frac {4}{3} k_z^2 {\mu} T \right ) \bar w + F^\prime \left .\frac {\partial \bar w}{\partial \hat {x}}\right \vert _\eta - \left (\frac {F}{2\hat {x}} + \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right )\frac {\partial \bar w}{\partial \eta } - \left (\frac {{\mu} }{2\hat {x} T} \frac {\partial \bar w}{\partial \eta }\right )^\prime \nonumber \\ & + \left (\frac {k_z^2{\mu} ^\prime T^\prime F}{3\hat {x}} + \frac {2k_z^2{\mu} ^\prime T^\prime }{3}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta - \frac {2k_z^2{\mu} ^\prime F^\prime }{3}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta \right ) \bar \tau - k_z^2 T \bar p = 0, \end{align}
\begin{align} \text{$\left .{\mathcal{E}}\right \vert $ }& \left (\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta - \frac {\eta _cT^\prime }{2\hat {x}}\right )\bar u -\left (\gamma -1\right ){\textit{Ma}}^2\frac {{\mu} F^{\prime \prime }}{\hat {x} T}\frac {\partial \bar u}{\partial \eta } + \frac {T^\prime }{T}\bar v + \left [\frac {FT^\prime }{2\hat {x} T} - {\textrm{i}}{\mathcal{F}} - \frac {F^\prime }{T}\left .\frac {\partial T}{\partial \hat {x}}\right \vert _\eta \right . \nonumber \\ & \left . + \frac {T^\prime }{T}\left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta - \left (\gamma -1\right ){\textit{Ma}}^2\frac {{\mu} ^\prime (F^{\prime \prime })^2}{2\hat {x} T} + \frac {k_z^2{\mu} T}{\textit{Pr}}\right ]\bar \tau - \frac {1}{\textit{Pr}}\left (\frac {{\mu} ^\prime T^\prime }{2\hat {x} T} \bar \tau \right )^\prime + F^\prime \left .\frac {\partial \bar \tau }{\partial \hat {x}}\right \vert _\eta \nonumber \\ &- \left ( \frac {F}{2\hat {x}} + \left .\frac {\partial F}{\partial \hat {x}}\right \vert _\eta \right ) \frac {\partial \bar \tau }{\partial \eta } - \frac {1}{\textit{Pr}}\left (\frac {{\mu} }{2\hat {x} T}\frac {\partial \bar \tau }{\partial \eta }\right )^\prime = 0, \end{align}
where
\begin{equation} {G}=\frac {{\textit{Re}}^2}{r_c} = \left (\frac {\rho _\infty ^\ast U_\infty ^\ast }{{\mu} _\infty ^\ast }\right )^2 \frac {\lambda _z^{\ast 3}}{8\pi ^3 r_c^\ast } = O(1) \end{equation}
is the global Görtler number (Kobayashi Reference Kobayashi1972; Wu et al. Reference Wu, Zhao and Luo2011) and
${\mu} ^\prime \equiv {\textrm{d}}{\mu} /{\textrm{d}} T$
. The global Görtler number (2.16) is defined using
$\lambda _z^\ast$
as the reference length, whereas the local Görtler number
$G_\delta = (\rho _\infty ^\ast U_\infty ^\ast \delta _0^\ast /{\mu} _\infty ^\ast )(\delta _0^\ast /r_c^\ast )^{1/2}$
(e.g. Floryan & Saric Reference Floryan and Saric1979; El-Hady & Verma Reference El-Hady and Verma1984) is based on the boundary-layer thickness
$\delta ^\ast$
at a specified location
$x_0^\ast$
. The two are related by
$G = G_\delta ^2 (\varLambda ^\ast /\delta _0^\ast )^3$
. The global Görtler number arises from the asymptotic scaling (2.14) and the use of
$\lambda _z^\ast$
as reference length. It distils the effect of constant disturbance spanwise wavelength and wall curvature and is independent of the growth of the laminar base flow with and without suction (Kobayashi Reference Kobayashi1972; Wu et al. Reference Wu, Zhao and Luo2011). Equations (2.15a
)–(2.15e
) are parabolic in
$\hat {x}$
and represent an initial-boundary value problem that needs to be solved subject to appropriate initial and boundary conditions.
2.2.1. Wall boundary conditions
The no-slip and no-penetration conditions
$\bar u=\bar v=\bar w=0$
are assumed over an adiabatic wall,
$\partial \bar \tau /\partial \eta =0$
(El-Hady & Verma Reference El-Hady and Verma1984; Viaro & Ricco Reference Viaro and Ricco2019a
). The use of the no-penetration condition
$\bar v=0$
at the wall is justified as follows. In a laboratory, distributed suction is usually applied through a porous substrate where
$\bar v$
and
$\bar p$
are coupled via a Darcy law (e.g. Wedin, Cherubini & Bottaro Reference Wedin, Cherubini and Bottaro2015)
\begin{equation} v^\ast = - \frac {K^\ast }{{\mu} ^\ast } \frac {\partial p^\ast }{\partial y^\ast }, \end{equation}
where
$K^\ast \,[{{\textrm {m}}^2}]$
is the permeability. In non-dimensional form, relation (2.17) at the wall becomes
\begin{equation} \grave v\lvert _{y=0} = \left . - \frac {{Da}\,{\textit{Re}}}{{\mu} _w}\frac {\partial \grave p}{\partial y}\right \lvert _{y=0}, \end{equation}
where
${\mu} _w={\mu} (T_w)=O(1)$
is the dynamic viscosity at the wall, the Darcy number
${Da} \equiv K^\ast /\varLambda ^{\ast 2}\ll 1$
is typically
$O({\textit{Re}}^{-1})$
or smaller and
${Da}{\textit{Re}} = O(1)$
. Introducing the asymptotic scaling (2.14) in (2.18) leads to
$\bar v (\partial \bar p/\partial \eta )^{-1} = O({Da})$
, i.e. the coupling is inhibited, thus justifying the no-penetration condition at the wall.
Suction non-uniformities of small amplitude, such as those examined via linear stability analysis by Floryan (Reference Floryan1997), Roberts et al. (Reference Roberts, Floryan, Casalis and Arnal2001) and Roberts & Floryan (Reference Roberts and Floryan2002, Reference Roberts and Floryan2008) may alter the stability of TS waves and generate unstable streamwise vortices. The latter are not attenuated by increasing the mean suction rate. For incompressible boundary-layer flows, suction non-uniformities proportional to
$\cos (\tilde \alpha x)$
with streamwise wavenumbers
$\tilde \alpha$
in the unstable band
$0.2\lt \tilde \alpha \lt 3$
were considered. These instabilities would not occur in boundary layers over suction porous sets relevant to the present work because the pore size and spacing are
$d_p^\ast \ll \lambda _z^\ast$
. The corresponding suction wavenumber,
$\tilde \alpha = 2\pi \varLambda ^\ast /d_p^\ast \gg 1$
, would therefore lie well outside the unstable range. The destabilising effect of non-uniformity at the pore scale was not observed in the wind tunnel experiments of Fransson & Alfredsson (Reference Fransson and Alfredsson2003), Yoshioka et al. (Reference Yoshioka, Fransson and Alfredsson2004) and Kurian & Fransson (Reference Kurian and Fransson2011).
Wall roughness can also be neglected because the roughness height is assumed to be comparable to the pore size
$d_p^\ast$
and, therefore, to be much smaller than the boundary-layer thickness
$\delta ^\ast$
. The use of the no-slip conditions,
$\bar u=\bar w=0$
, is thus justified.
Periodic spanwise modulations of wall velocity or temperature produce finite commensurate or infinite incommensurate states, depending on the ratio of the disturbance wavenumber
$k_z$
to that of the wall modulation (Panday & Floryan Reference Panday and Floryan2023). Irrational ratios yield infinitely many aperiodic states, which cannot be investigated through DNSs, linear stability analysis or the present asymptotic approach due to the requirement of an infinitely wide computational domain. We therefore focus on the linear growth of a single monochromatic disturbance with constant
$\lambda _z^\ast$
, assuming a uniform suction velocity for the base flow and zero disturbance velocity at the wall.
These assumptions are consistent with the experiments of Fransson & Alfredsson (Reference Fransson and Alfredsson2003) who used a sintered-plastic porous layer with a permeability of
$K^\ast = 3.7\,{{{\rm \unicode{x03BC}} \textrm {m}}^2}$
, an average pore size of
$d_p^\ast = 16\,{\rm \unicode{x03BC}} {\textrm {m}}$
and a roughness height of
$0.38\,{\rm \unicode{x03BC}} {\textrm {m}}$
. Such a low roughness is typical of polished, non-permeable surfaces used in supersonic quiet tunnels (Schneider Reference Schneider2008a
).
2.2.2. Outer boundary conditions
Both
$\bar u$
and
$\bar \tau$
vanish in the free stream (
$\eta \gg 1$
), while
$\bar v$
,
$\bar w$
and
$\bar p$
satisfy boundary conditions of the mixed type
\begin{align} \frac {\partial \bar v}{\partial \eta } + \left \vert k_z\right \vert (2\hat {x})^{1/2} \bar v &\to -{\textrm{e}}^{{\textrm{i}}{\mathcal{F}}\hat {x} + {\textrm{i}} k_y(2\hat {x})^{1/2}\left [\eta -\beta _c\left (\hat {x}\right )\right ] - \big(k_y^2+k_z^2\big)\hat {x}}, \end{align}
\begin{align} \frac {\partial \bar w}{\partial \eta } + \left \vert k_z\right \vert (2\hat {x})^{1/2} \bar w &\to {\textrm{i}} k_y (2\hat {x})^{1/2} {\textrm{e}}^{{\textrm{i}}{\mathcal{F}}\hat {x} + {\textrm{i}} k_y (2\hat {x})^{1/2}\left [\eta -\beta _c\left (\hat {x}\right )\right ] - \big(k_y^2+k_z^2\big)\hat {x}}, \end{align}
\begin{align} \frac {\partial \bar p}{\partial \eta } + \left \vert k_z\right \vert \big(2\hat {x}\big)^{1/2} \bar p &\to 0, \end{align}
which differ from the those used by LWG99, Ricco & Wu (Reference Ricco and Wu2007) and Viaro & Ricco (Reference Viaro and Ricco2019a
) because the term
$\beta _c (\hat {x} ) = \lim _{\eta \to \infty } [\eta - F (\hat {x},\eta ) ]$
is streamwise-dependent.
The outer boundary conditions (2.19a
) are derived by matching the outer limit of region III (i.e. the large-
$\eta$
limit of the CLUBR equations (2.15a
)–(2.15e
)) with the inner limit of region IV. Following Wundrow & Goldstein (Reference Wundrow and Goldstein2001), the flow in region IV – where
$\hat {x},\hat {y}=O(1)$
and
$\hat {y}={\textit{Re}}^{-1}y$
– is decomposed in its steady two-dimensional and unsteady three-dimensional parts at
$O(\epsilon )$
:
\begin{align} \boldsymbol{u} &= \left \lbrace 1,0,0\right \rbrace + \epsilon \lbrace \acute u, \acute v, 0 \rbrace (\hat {x},\hat {y}) + \epsilon \lbrace \grave u, \grave v, \grave w \rbrace (\hat {x},y,\hat {y},z,\hat {t}) + \ldots , \end{align}
\begin{align} p &= \frac {1}{\gamma {\textit{Ma}}^2} + \epsilon \acute p (\hat {x},\hat {y}) + \epsilon ^2 \grave p (\hat {x},y,\hat {y},z,\hat {t}) + \ldots, \end{align}
\begin{align} \tau &= 1 + \epsilon \acute \tau (\hat {x},\hat {y}) + \epsilon \grave \tau (\hat {x},y,\hat {y},z,\hat {t}) + \ldots \, . \end{align}
By introducing the expansion (2.20a
) in the Navier–Stokes and perfect gas equations, one finds a system of equations for the steady, two-dimensional and the unsteady, three-dimensional part of the flow in region IV. The steady two-dimensional part
$\lbrace 1,0,0\rbrace + \epsilon \lbrace \acute u, \acute v, 0 \rbrace (\hat {x},\hat {y})$
is induced by the base-flow displacement and is governed by the steady Euler equations. Its wall-normal velocity component
$\epsilon \acute v$
matches the base flow velocity
$V$
(2.11b
) when
$r_t\ll 1$
,
\begin{equation} \lim _{\hat {y}\to 0}\epsilon \acute v = \lim _{\eta \to \infty }V = \frac {1}{{\textit{Re}}}\frac {{\textrm{d}}}{{\textrm{d}}\hat {x}}\big[\big(2\hat {x}\big)^{1/2}\beta _c\big(\hat {x}\big)\big] + \frac {1}{{\textit{Re}}}\frac {\gamma _c (\hat {x})}{(2\hat {x})^{1/2}}, \end{equation}
where
$\gamma _c (\hat {x} ) = \lim _{\eta \to \infty } (\eta _c-\eta )$
. The stream function for the first two terms in (2.20a
) is
$\varPsi (\hat {x},y,\hat {y} ) = y + \psi _1 (\hat {x},\hat {y} )$
, where
\begin{equation} \epsilon \acute v = - \frac {1}{{\textit{Re}}}\frac {\partial \psi _1}{\partial \hat {x}} = - \frac {1}{{\textit{Re}}}\frac {\partial \varPsi }{\partial \hat {x}}. \end{equation}
Equating (2.21) and (2.22) and integrating yields
\begin{equation} \varPsi (\hat {x},y) = y - (2\hat {x})^{1/2}\beta _c(\hat {x}) - \int ^{\hat {x}}\frac {\gamma _c (\hat {x})}{(2\hat {x})^{1/2}}{\textrm{d}}\breve x \mbox{, for } \hat {y}\ll 1\mbox{, } y=O(1). \end{equation}
The stream function
$\psi _1$
valid for all
$\hat {x}$
and
$\hat {y}$
is obtained by solving a Laplace equation (in the subsonic case) or a wave equation (in the supersonic case). More details on the derivation of the equation for
$\psi _1$
are in Appendix A. Note that an analytical expression for
$\psi _1 (\hat {x},\hat {y} )$
cannot be obtained for all
$\hat {y}$
in the non-similar case or when the self-similar base flow is supersonic. However, the outer boundary conditions (2.19a
) are derived by matching the large-
$\eta$
limit of the CLUBR equations (2.15a
)–(2.15e
) with the small-
$\hat {y}$
limit of
$\lbrace \grave u, \grave v, \grave w \rbrace$
in (2.20a
) (LWG99). Hence, the focus is on the asymptotic behaviour of
$\psi _1$
as
$\hat {y} \to 0$
(i.e.
$\hat {y} \ll 1$
) given in (2.23).
The unsteady, three-dimensional part of (2.20) is governed by
\begin{align} \qquad\qquad\qquad & \left .{\mathcal{C}}\right \vert \frac {\partial \grave v}{\partial y} + \frac {\partial \grave w}{\partial z} = 0, \end{align}
\begin{align} \left .{\mathcal{Y}}\right \vert \frac {\partial \grave v}{\partial \hat {t}} & + \frac {\partial \grave v}{\partial \hat {x}} + r_t \left [(\acute v + \grave v)\frac {\partial \grave v}{\partial y} + \grave w \frac {\partial \grave v}{\partial z} + \frac {\partial \grave p}{\partial y}\right ] = \frac {\partial ^2\grave v}{\partial y^2} + \frac {\partial ^2\grave v}{\partial z^2} , \end{align}
\begin{align} \left .{\mathcal{Z}}\right \vert \frac {\partial \grave w}{\partial \hat {t}} & + \frac {\partial \grave w}{\partial \hat {x}} + r_t \left [(\acute v + \grave v)\frac {\partial \grave w}{\partial y} + \grave w \frac {\partial \grave w}{\partial z} + \frac {\partial \grave p}{\partial z}\right ] = \frac {\partial ^2\grave w}{\partial y^2} + \frac {\partial ^2\grave w}{\partial z^2} , \end{align}
$\grave u=\grave \tau =0$
and
$\grave \rho =0$
because of the perfect gas equation. The system (2.24) is elliptic in the cross-flow (
$y$
–
$z$
) plane and the viscous three-dimensional components are coupled to the inviscid two-dimensional ones because
$\acute v$
appears in (2.24). The velocity components are decoupled by substituting (2.22) and (2.23) into (2.24), which yields the linearised momentum balances valid for
$\hat {y}\ll 1$
(LWG99):
\begin{equation} \left (\frac {\partial }{\partial \hat {t}} + \left .\frac {\partial }{\partial \hat {x}}\right \vert _{\varPsi } - \frac {\partial ^2}{\partial \varPsi ^2} - \frac {\partial ^2}{\partial z^2}\right )\lbrace \grave v,\grave w \rbrace = 0. \end{equation}
The solution to (2.25) is
\begin{equation} \lbrace \grave v,\grave w \rbrace (\hat {x},\varPsi ,z,\hat {t}) = \lbrace \hat v,\hat w \rbrace (\hat {x}) {\textrm{e}}^{- {\textrm{i}}{\mathcal{F}}\hat {t} + {\textrm{i}} k_y\varPsi + {\textrm{i}} k_zz} + \textrm{c.c.}, \end{equation}
where the amplitude is found by matching with (2.1) for
$\varPsi \sim y=O(1)$
and
$\hat {x}\ll 1$
:
\begin{equation} \lbrace \hat v,\hat w \rbrace (\hat {x}) = \lbrace \hat v^\infty ,\hat w^\infty \rbrace {\textrm{e}}^{{\textrm{i}}{\mathcal{F}}\hat {x} - \big(k_y^2+k_z^2\big)\hat {x}} . \end{equation}
As
$T\sim 1$
and
$\bar Y = O(y)$
for
$y,\eta \gg 1$
, the far-field variable for region III,
$y^{(0)}$
, is defined consistently with (2.23):
\begin{equation} y^{(0)}\equiv (2\hat {x}{\mathcal{F}})^{1/2} [\eta - \beta _c (\hat {x})] - {\mathcal{F}}^{1/2}\int ^{\hat {x}}\frac {\gamma _c (\hat {x})}{(2\hat {x})^{1/2}}{\textrm{d}}\breve x. \end{equation}
By applying the transformation
$(\hat {x},\eta )\to (\hat {x},y^{(0)})$
to the large-
$\eta$
form of (2.15a
)–(2.15e
), the far-field equations are recovered:
\begin{align} & \left .{\mathcal{C}}\right \vert (2\hat {x}{\mathcal{F}})^{1/2}\frac {\partial \bar v}{\partial y^{(0)}} + \bar w = 0, \end{align}
\begin{align} \left .{\mathcal{Y}}\right \vert \left (\frac {1}{2\hat {x}} + k_z^2 - {\textrm{i}} {\mathcal{F}}\right )\bar v + \left .\frac {\partial \bar v}{\partial \hat {x}}\right \vert _{y^{0)}} &- \frac {{\mathcal{F}}^{1/2}\gamma _c(\hat {x})}{(2\hat {x})^{1/2}}\frac {\partial \bar v}{\partial y^{(0)}} - {\mathcal{F}}\frac {\partial ^2\bar v}{\partial y^{(0)2}} + \frac {{\mathcal{F}}^{1/2}}{(2\hat {x})^{1/2}}\frac {\partial \bar p}{\partial y^{(0)}} = 0, \end{align}
\begin{align} \left .{\mathcal{Z}}\right \vert (k_z^2 - {\textrm{i}}{\mathcal{F}})\bar w + \left .\frac {\partial \bar w}{\partial \hat {x}}\right \vert _{y^{(0)}} &- \frac {{\mathcal{F}}^{1/2}\gamma _c(\hat {x})}{(2\hat {x})^{1/2}}\frac {\partial \bar w}{\partial y^{(0)}} - {\mathcal{F}}\frac {\partial ^2\bar w}{\partial y^{(0)2}} - k_z^2\bar p = 0. \end{align}
The solution (2.19) that matches with (2.1) for
$\hat {x} \ll 1$
,
$y^{(0)} = O(1)$
and (2.26) for
$\hat {x} = O(1)$
,
$y^{(0)} \gg 1$
is first derived for
$\bar p$
by combining the wall-normal derivative of (2.29b
) with (2.29c
) and using (2.29a
). The spanwise component
$\bar w$
is obtained by solving the inhomogeneous heat equation
\begin{equation} \left .\frac {\partial \bar w^0}{\partial \hat {x}}\right \vert _{\grave y^{(0)}} - {\mathcal{F}} \frac {\partial ^2\bar w^0}{\partial \grave y^{(0)2}} = k_z^2 \bar p {\textrm{e}}^{\left (k_z^2 - {\textrm{i}}{\mathcal{F}}\right )\hat {x}}, \end{equation}
where the change of variable
$\bar w = \bar w^0 {\textrm{e}}^{ ({\textrm{i}}{\mathcal{F}}-k_z^2 )\hat {x}}$
,
$\grave y^{(0)} = (2\hat {x}{\mathcal{F}} )^{1/2} [\eta - \beta _c (\hat {x} ) ]$
was used to simplify (2.29c
) (Polyanin & Nazaikinskii Reference Polyanin and Nazaikinskii2015, § 3.8.7.4). Equation (2.30) is solved with initial conditions derived by matching
$\bar {v}$
and
$\bar {w}$
to region I (2.1), while the outer conditions are established by matching
$\bar {v}$
and
$\bar {w}$
to the small-
$\hat {y}$
limit of region IV (2.26). The wall-normal component
$\bar {v}$
is derived by integrating (2.29a
).
2.2.3. Initial conditions
The initial conditions are imposed in the impermeable region at
$\hat {x}\ll 1$
and are therefore those of Viaro & Ricco (Reference Viaro and Ricco2019a
, appendix B). The evolution from the impermeable to the suction region occurs gradually across an adjustment strip of finite width
$\Delta \hat {x}_s$
(red area in figure 1) and centred at a prescribed
$\hat {x}_s$
. In this region, the suction velocity varies as
$V_wS(\hat {x})$
, where
\begin{equation} S (\hat {x}) = \begin{cases} 0, & \text{for $\hat {x} - \hat {x}_s \leqslant -\Delta \hat {x}_s/2$},\\ \left [ 1 + \exp \left (\displaystyle {\frac {1}{ \tilde x - 1 } + \frac {1}{ \tilde x}}\right ) \right ]^{-1}, & \text{for $\left \vert \hat {x} - \hat {x}_s\right \vert \leqslant \Delta \hat {x}_s/2$},\\ 1, & \text{for $\hat {x} - \hat {x}_s \gt \Delta \hat {x}_s/2$}, \end{cases} \end{equation}
and
$\tilde x = (\hat {x}-\hat {x}_s+\Delta \hat {x}_s/2 )/\Delta \hat {x}_s$
(Negi, Mishra & Skote Reference Negi, Mishra and Skote2015). We have verified that varying the length of the adjustment strip between
$\Delta \hat {x}_s=0.01$
and
$0.1$
has a negligible effect on the results. We assume
$\Delta \hat {x}_s = 0.05$
for all the computations. In the results section,
$\gamma _w$
indicates the constant suction value attained at the end of the adjustment strip.
2.2.4. Numerical procedures
The base flow and the CLUBR equations are solved by marching downstream as they are parabolic in
$\hat {x}$
. At each
$\hat {x}$
location, the solution is obtained by a second-order accurate block-elimination algorithm (Cebeci Reference Cebeci2002), also used to solve (2.13). When suction is applied, the boundary layer becomes thinner in
$\eta$
and thus a non-uniform grid is used to resolve the near-wall region. More details on the numerical method are found in Appendix B.
3. Results
The base-flow solution (2.5) is validated in § 3.1.1 against wind-tunnel measurements and DNSs of boundary layers with uniform suction. The free stream conditions outlined in table 1 are used in the rest of § 3. The characteristics of the porous wall and the base-flow properties are discussed in §§ 3.1.2 and 3.1.3, respectively. The combined effect of receptivity and uniform suction on the growth of Görtler vortices is studied in § 3.2. The neutral curves are examined in § 3.3.
Free stream conditions reported in wind-tunnel tests over impermeable surfaces. Cited studies are Flechner et al. (Reference Flechner, Jacobs and Whitcomb1976, F76), Graziosi & Brown (Reference Graziosi and Brown2002, GB02), Maslov et al. (Reference Maslov, Shiplyuk, Sidorenko and Arnal2001, M01), Running et al. (Reference Running, Bemis, Hill, Borg, Redmond, Jantze and Scalo2023, R23), Tani (Reference Tani1962, T62), Finnis & Brown (1997, FB97), Ciolkosz & Spina (Reference Ciolkosz and Spina2006, CS06), Wang et al. (Reference Wang, Wang and Zhao2018, W18), Li et al. (Reference Li, Choudhari, Paredes, Schneider and Portoni2018, L18) and de Luca et al. (Reference de Luca, Cardone, Aymer de la Chevalerie and Fonteneau1993, L93). The flow conditions used in §§ 3.2 and 3.3 are highlighted in pink. The subscript
$o$
denotes the stagnation quantities.
(a,b) Laminar velocity profiles in the suction region (solid curves) compared with the incompressible data (marks) of (a) Fransson & Alfredsson (Reference Fransson and Alfredsson2003) (
$x_s^\ast =0.34\,\textrm {m}$
) and (b) Kay (Reference Kay1953) (
$x_s^\ast =0.11\,\textrm {m}$
). Asymptotic-suction profiles are plotted in bold black. (c) Momentum thickness along the surface of a cone at Mach 6 and the numerical data of Hollender et al. (Reference Hollender, Dwivedi and Candler2019) (
$x_s^\ast = 0.2\,\textrm {m}$
).
3.1. Laminar base flow
3.1.1. Comparison with experiments and DNSs
The non-similar laminar base flow (2.5) with wall suction is validated against the low-speed wind tunnel data of Fransson & Alfredsson (Reference Fransson and Alfredsson2003) (figure 3
a). The free stream velocity is
$U_\infty ^\ast = 5\,\textrm {m}\,\textrm {s}^{-1}$
and a region of uniform suction velocity
$V^\ast (\hat {x},0 ) = -1.44\,\textrm {cm}\,\textrm {s}^{-1}$
begins at
$x_{s}^\ast = 0.36\,\textrm {m}$
. We assume a spanwise wavelength
$\lambda _z^\ast =\delta _{99}^\ast (x_{s}^\ast ) = 5.2\,\textrm {mm}$
, which yields
${\textit{Re}} = 275$
and
$\gamma _w = - 0.79$
. The results of (2.5) (figure 3
a) show excellent agreement with the wind tunnel data. At
$x^\ast = 0.9\,\textrm {m}$
, the boundary layer reaches the asymptotic-suction state, denoted by the black bold curve of the incompressible asymptotic-suction profile
$U = 1 - \exp (\gamma _w y)$
. Fransson & Alfredsson (Reference Fransson and Alfredsson2003) had previously validated their results by solving a non-similar incompressible boundary-layer equation with a sharp transition from the impermeable to the suction region, i.e. without the smooth adjustment strip described in § 2.2.3 (
$\Delta \hat {x}_s = 0$
). Their mathematical formulation differs in that it features the streamwise variable
$\xi = \gamma _w \hat {x}^{1/2}$
and the scalings
$F(\xi ,\eta ) = \psi (\hat {x},y) \gamma _w\xi ^{-1}$
and
$\eta = y \gamma _w\xi ^{-1}$
, all dependent on the suction rate
$\gamma _w$
.
A good agreement is also found with the wind-tunnel data of Kay (Reference Kay1953) (figure 3
b). They studied an incompressible boundary layer with
$U_\infty ^\ast = 17.37\,\textrm {m}\textrm {s}^{-1}$
and developing over a porous surface. The boundary-layer thickness was
$\delta _{99}^\ast = 1.5\textrm {mm}$
at the beginning of the suction region, located at
$x_{s}^\ast = 0.11\textrm {m}$
, the kinematic viscosity was
$\nu ^\ast = 1.43 \times 10^{-5}\,{\textrm {m}}^2\,{\textrm {s}^{-1}}$
and the Reynolds number based on
$\delta _{99}^\ast$
was 290. A uniform suction velocity
$V (\hat {x},0 ) = - 5.1\,\textrm {cm}\,\textrm {s}^{-1}$
was applied and the laminar asymptotic suction profile was attained at
$x^\ast = 0.35\,\textrm {m}$
.
The velocity and temperature profiles of compressible laminar boundary layers over porous plates with suction are not readily available in the experimental and numerical literature. The wall-normal profiles reported in Leontiev & Pavlyuchenko (Reference Leontiev and Pavlyuchenko2008) at
${\textit{Ma}}=2.5$
cannot be used for comparison because they were measured in a high-intermittency region downstream of a perforated suction plate, where the flow was already transitional. Hollender et al. (Reference Hollender, Dwivedi and Candler2019) performed DNSs on the cone-flared cone geometry studied by Li et al. (Reference Li, Choudhari, Paredes, Schneider and Portoni2018). Unfortunately, we are unable to validate our computations against the velocity profile they sampled in the suction region as neither the corresponding boundary-layer thickness nor the exact location of the suction region on the cone was specified. Nevertheless, Hollender et al. (Reference Hollender, Dwivedi and Candler2019) reported the distribution of the momentum thickness upstream of and over the suction region. They imposed a constant mass flow rate
$\dot {m}_{\textit{sur}}^\ast =-0.5\,\textrm {g}\textrm {s}^{-1}$
on the surface of a truncated section of a cone with a half-angle
$\alpha _{\textit{cone}}=7^{\circ }$
. The fore section of the model was conical and had a slant height of
$0.3\,\textrm {m}$
. The suction region was located between
$\ell ^\ast _1=0.20\,\textrm {m}$
and
$\ell ^\ast _2=0.28\,\textrm {m}$
from the tip and covered an estimated area
\begin{equation} A^\ast _{\textit{suc}} = \pi \left (\ell _1^\ast +\ell _2^\ast \right )\sin \left (\alpha _{\textit{cone}}\right ) \left (\ell _2^\ast -\ell _1^\ast \right ) = 0.0147\,{\textrm {m}^2} . \end{equation}
The free stream density and temperature were
$\rho _\infty ^\ast = 0.035\,{\textrm {kg}\,\textrm {m}^{-3}}$
and
$T_\infty ^\ast =52.6\,{\textrm {K}}$
. The boundary layer on the cone at zero angle of attack develops downstream of an axisymmetric conical shock wave, with no fluid motion occurring along the meridian planes. The inviscid flow properties at the cone surface are described by the Taylor–Maccoll theory and the boundary layer is governed by the compressible Blasius solution (Stewartson Reference Stewartson1964). For
${\textit{Ma}}=6$
and
$\alpha _{\textit{cone}} = 7.5^{\circ }$
, the tables in Sims (Reference Sims1964) report the following thermodynamic properties at the cone surface:
\begin{align} \begin{array}{cccc} \displaystyle \frac {p_{\textit{sur}}^\ast }{p_\infty ^\ast } = 2.08, & \displaystyle \frac {\rho _{\textit{sur}}^\ast }{\rho _\infty ^\ast } = 1.68, & \displaystyle \frac {T_{\textit{sur}}^\ast }{T_\infty ^\ast } = 1.24, & {\textit{Ma}}_{\textit{sur}} = 5.3. \end{array} \end{align}
The inviscid surface velocity is
$U_{\textit{sur}}^\ast ={\textit{Ma}}_{\textit{sur}}\sqrt {\gamma {\mathcal{R}}^\ast T_\infty ^\ast }= 857.53\,\textrm {m}\textrm {s}^{-1}$
. Assuming
$\lambda _z^\ast = \delta _{99}^\ast = 1\,\textrm {mm}$
(Li et al. Reference Li, Choudhari, Paredes, Schneider and Portoni2018), the Reynolds number is
${\textit{Re}}_{\textit{sur}} = \rho _{\textit{sur}}^\ast U_{\textit{sur}}^\ast \lambda _z^\ast /(2\pi {\mu} _{\textit{sur}}^\ast )$
, where the viscosity at the surface is estimated using Sutherland’s law
\begin{equation} \frac {{\mu} ^\ast }{{\mu} _{\textit{ref}}^\ast } = \left (\frac {T^\ast }{T_{\textit{ref}}}\right )^{3/2}\frac {\chi ^\ast + T_{\textit{ref}}^\ast }{\chi ^\ast + T^\ast }, \end{equation}
with
${\mu} _{\textit{ref}}^\ast = 1.85\boldsymbol{\cdot }10^{-5}\,\textrm {Pa}\,\textrm {s}$
,
$T_{\textit{ref}}^\ast =300\,{\textrm {K}}$
,
$\chi ^\ast = 129\,{\textrm {K}}$
. The adiabatic recovery temperature is (Anderson Reference Anderson2016)
\begin{equation} T_{w}^\ast = T_{\textit{sur}}^\ast \left (1+\frac {\gamma -1}{2}Pr^{1/2}{\textit{Ma}}_{\textit{sur}}^2\right ) = 374\,{\textrm {K}}. \end{equation}
The wall-normal velocity is found from the mass flow rate
$\dot m_{w}^\ast$
:
\begin{equation} V_w^\ast = \frac {\dot {m}_{w}^\ast }{\rho _w^\ast A_{\textit{suc}}^\ast } = \frac {\dot {m}_{w}^\ast }{A_{\textit{suc}}^\ast } \frac {T_w^\ast }{\rho _{\textit{sur}}^\ast T_{\textit{sur}}^\ast } . \end{equation}
The surface temperature of the cone is
$T_w^\ast =300\,{\textrm {K}}$
and the suction parameter is
\begin{equation} \gamma _w = \frac {\dot {m}_{w}^\ast }{A_{\textit{suc}}^\ast } \frac {T_w^\ast }{T_{\textit{sur}}^\ast } \frac {\lambda _z^\ast }{2\pi {\mu} _{\textit{sur}}^\ast } = - 6.01. \end{equation}
The boundary-layer momentum thickness is computed from our base-flow solution (2.5),
\begin{equation} \theta _{BL} = \int _0^\infty \frac {U}{T}\left (1-U\right ) {\textrm{d}} y, \end{equation}
and agrees well with the DNS results, as shown in figure 3(c).
3.1.2. Suction rates through the porous wall
The results presented from this point onwards are computed using the free stream conditions of cases F76 and W18 in table 1. Since suction is typically achieved by generating a pressure gradient across a porous membrane, the implementation of a transpiration surface is constrained by the low static pressures in supersonic wind tunnels. Therefore, one might question whether the values of
$\gamma _w$
considered herein are feasible in practice. Although the design of suction actuators is beyond the scope of this work, we provide an estimate of the membrane’s thickness in this section. The velocity through the porous layer is assumed to be uniform and equal to
$V_w$
and the flow is governed by the linear Darcy law (2.17) (Schmidt et al. Reference Schmidt, Bitter, Hornung and Shepherd2016; Traub et al. Reference Traub, Kube, Jose, Prasannakumar and Hühne2024). The use of (2.17) is justified when the pore-scale Reynolds number is sufficiently small
\begin{equation} {\textit{Re}}_d \equiv \frac {\rho _w^\ast V_w^\ast d_p^\ast }{{\mu} _w^\ast } = \frac {p_\infty ^\ast }{{\mathcal{R}}^\ast T_w^\ast } \frac {\gamma _w U_\infty ^\ast }{{\textit{Re}}} \frac {d_p^\ast }{{\mu} _w^\ast } \ll 1. \end{equation}
In (3.8),
$T_w^\ast$
is computed using (3.4) with
$T_\infty ^\ast$
and
$\textit{Ma}$
instead of
$T_{\textit{sur}}^\ast$
and
${\textit{Ma}}_{\textit{sur}}$
. The effect of choking, which may be important at high suction rates (Li et al. Reference Li, Choudhari, Paredes, Schneider and Portoni2018), is not considered (Schmidt et al. Reference Schmidt, Bitter, Hornung and Shepherd2016). The pore diameter and permeability of Fransson & Alfredsson (Reference Fransson and Alfredsson2003),
$d_p^\ast =16\,{\rm \unicode{x03BC}} {\textrm {m}}$
and
$K^\ast = 3.7 \times 10^{-12}\,{\textrm {m}^2}$
, are used. The combination of low pressure and high recovery temperature yields a relatively high mean-free path
$\ell _{mfp}^\ast = (\pi {\mathcal{R}}^\ast T_w^\ast /2)^{1/2} {\mu} _w^\ast /p_\infty ^\ast$
and Knudsen number based on the pore diameter
${Kn}=\ell _{mfp}^\ast /d_p^\ast$
. The flow within the pores lies in the slip regime (
$0.001\lt {Kn}\leqslant 0.1$
) and the permeability is enhanced by a factor
$1+7.23{Kn}$
(Yang & Weigand Reference Yang and Weigand2018). The theoretical maximum pressure difference across a porous layer of thickness
$\Delta h^\ast$
is the static pressure in the free stream,
$\Delta p^\ast = p_\infty ^\ast$
, if one assumes that a steady vacuum is generated underneath. Assuming
$\partial p^\ast /\partial y^\ast \cong \Delta p^\ast /\Delta h^\ast$
, the Darcy law (2.17) becomes
\begin{equation} \frac {\Delta h^\ast }{p_\infty ^\ast } = - \frac {K^\ast \left (1+7.23{Kn}\right )}{{\mu} _w^\ast }\frac {{\textit{Re}}}{\gamma _w U_\infty ^\ast }, \end{equation}
where
$\Delta h^\ast$
represents the maximum theoretical thickness of the porous layer through which a given suction rate
$\gamma _w$
can be realised assuming that a vacuum is generated underneath.
The estimated maximum thicknesses pertinent to the considered range of
$\gamma _w$
are shown in the rightmost column in table 2 along with the pore-scale Reynolds number and the Knudsen number. In all cases,
$\textit{Kn}$
is within the limits of the slip regime and
${\textit{Re}}_d$
is much smaller than unity. The value of
$\Delta h^\ast$
decreases dramatically in the supersonic case due to the higher
$\gamma _w$
, higher
$T_w^\ast$
and lower
$p_\infty ^\ast$
. The smallest thickness
$\Delta h^\ast =348\,{\rm \unicode{x03BC}} {\textrm {m}}$
corresponds to the case
${\textit{Ma}} = 2.95$
and
$\gamma _w=-10$
, and is approximately 21 times
$d_p^\ast$
.
Properties of the flow within the porous wall from Flechner et al. (Reference Flechner, Jacobs and Whitcomb1976, F76) and Wang et al. (Reference Wang, Wang and Zhao2018, W18).
3.1.3. Compressible boundary layers
Effect of uniform suction on the velocity and temperature base-flow profiles at
${\textit{Ma}}=0.80$
and for different
$\hat {x}$
. The no-suction cases (self-similar base flow) are plotted in grey for comparison. The evolution of the velocity
$U$
(a,b) and temperature
$T$
(c,d) profiles is shown for
$\gamma _w=-2$
(a,c) and
$\gamma _w=-5$
(b,d). The asymptotic suction profiles are plotted with the circles.
Effect of uniform suction on the velocity and temperature base-flow profiles at
${\textit{Ma}}=2.95$
and for different
$\hat {x}$
. The no-suction cases (self-similar base flow) are plotted in grey for comparison. The evolution of the velocity
$U$
(a,b) and temperature
$T$
(c,d) profiles is shown for
$\gamma _w=-5$
(a,c) and
$\gamma _w=-10$
(b,d). The asymptotic suction profiles are plotted with the circles.
The streamwise evolution of the base flow profiles at
${\textit{Ma}}=0.80$
(F76) and
${\textit{Ma}}=2.95$
(W18) are shown in figures 4 and 5, respectively. As the wall is adiabatic and
$\hat {x}_s$
and
$\Delta \hat {x}_s$
are fixed, the behaviour of the laminar base flow is determined by
$\gamma _w$
and
$\textit{Ma}$
. The ASBL profiles (red circles) are obtained by solving the system (2.13). The profiles of
$U$
and
$T$
shown in figures 4 and 5 are sampled at
$\hat {x}=0.1$
and
$\hat {x}=0.3$
and represent the suction (red) and no-suction (grey) cases. While the velocity and temperature profiles without suction become thicker in
$y$
when suction is absent and the solution is self-similar, they remain close to the wall and adjust to the ASBL when suction is applied.
When suction is mild, such as for
$\gamma _w=-2$
at
${\textit{Ma}}=0.80$
(figure 4
a,c) or
$\gamma _w=-5$
at
${\textit{Ma}}=2.95$
(figure 5
b,d), the boundary-layer thickens over the suction region before reaching the ASBL state. Moreover, for these values of
$\gamma _w$
and
$\textit{Ma}$
, the temperature adjusts to the ASBL conditions more gradually than the velocity. The opposite occurs for intense suction, such as for
$\gamma _w=-5$
at
${\textit{Ma}}=0.80$
(figure 4
a,c) or
$\gamma _w=-10$
at
${\textit{Ma}}=2.95$
(figure 5
b,d), in which cases the boundary-layer thickness decreases downstream of
$\hat {x}_s$
. A thinning boundary layer is also observed in the DNSs of Hollender et al. (Reference Hollender, Dwivedi and Candler2019) (figure 3
c). The streamwise distance along which the Blasius profile shifts to the asymptotic-suction profile becomes shorter as
$\gamma _w$
increases and
$\textit{Ma}$
is fixed. Even when
$\gamma _w$
is intense, the streamwise-evolving temperature profile and the ASBL temperature profile still differ slightly near the wall, even though the streamwise velocity profiles show excellent collapse (insets in figures 4
d and 5
d).
For a given
$\gamma _w$
, the effect of suction weakens as the Mach number and the wall temperature increase. For instance, the same suction rate
$\gamma _w=-5$
produces a slower adjustment at
${\textit{Ma}}=2.95$
than at
${\textit{Ma}}=0.80$
from the Blasius boundary layer to the ASBL. A higher wall temperature
$T_w=T(\hat {x},0)$
decreases the amplitude of the forcing term in the boundary condition (2.12), and a given
$V_w$
generates a lower mass flow rate through the wall when
$\textit{Ma}$
is higher.
3.2. Attenuation of compressible Görtler vortices
Uniform suction: effect of
$\gamma _w$
and
$\lambda _z^\ast$
on the maximum of the streamwise-velocity disturbances (a,b) and the temperature disturbances (c,d) in steady conditions at
${\textit{Ma}}=0.8$
(a,c) and
${\textit{Ma}}=2.95$
(b,d). The length of the impermeable region is
$x_s^\ast =4.2\,\textrm {cm}$
in the subsonic case and
$x_s^\ast =2.5\,\textrm {cm}$
in the supersonic case. The magenta and orange curves show the cases
$\lambda _z^\ast =1\,\textrm {mm}$
and
$2\,\textrm {mm}$
, respectively.
The effect of suction on the receptivity to steady free stream vorticity is examined. In this section, the geometry is fixed. We consider a typical
$r_c^\ast =5\,\textrm {m}$
for the subsonic case (Viaro & Ricco Reference Viaro and Ricco2019a
) and take
$r_c^\ast =1\,\textrm {m}$
as in the experiments of W18 for the supersonic case. The free stream conditions of F76 and W18 are adopted (with fixed
$k_y=1$
) and we study the effect of different
$\gamma _w$
and
$\lambda _z^\ast$
on the boundary-layer perturbations. For given free stream conditions and wall curvature, varying
$\gamma _w$
impacts the base flow only, while varying
$\lambda _z^\ast$
changes
${\textit{Re}}$
and
$G$
(the latter grows with the third power of
$\lambda _z^\ast$
, as given in (2.16)).
The evolution of the amplitudes of the streamwise velocity and temperature perturbations is shown in figure 6 for
$\lambda _z^\ast =1\,\textrm {mm}$
(magenta) and
$2\,\textrm {mm}$
(orange) as a function of
$x^\ast /x_s^\ast$
. Suction markedly attenuates the growth of the streamwise velocity and temperature fluctuations (
$\vert \bar {u} \vert$
and
$\vert \bar {\tau } \vert$
are multiplied by
${\textit{Re}}$
in figure 6 to study quantities that are proportional to the physical variables in (2.14) as
$\lambda _z^*$
varies). The attenuation effect of suction intensifies as
$\gamma _w$
increases, although it weakens as the conditions change from subsonic to supersonic, in qualitative agreement with the results of linear stability theory (El-Hady & Verma Reference El-Hady and Verma1984). In the high-subsonic case (figure 6
a,c), a suction rate of
$\gamma _w=-5$
turns exponentially growing Görtler vortices with
$\lambda _z^\ast =1\,\textrm {mm}$
into decaying laminar streaks. In the supersonic case, an intense suction rate of
$\gamma _w=-10$
renders the perturbations constant along the streamwise direction. A transient behaviour is observed around
$x_s^\ast$
in the supersonic case for
$\gamma _w=-10$
(refer to inset in figure 6
b). The disturbance amplitude rapidly decreases, reaches a minimum downstream and then increases again under the forcing exerted by free stream vorticity.
In the absence of suction, increasing
$\lambda _z^\ast$
delays the onset of Görtler vortices. The delay occurs because the condition for large centrifugal growth is weakened, i.e.
$\delta ^*$
becomes comparable to
$\lambda _z^\ast$
farther downstream and thus the role of the
${\mathcal{Y}}$
-momentum balance, where the centrifugal effect is active, is postponed. The influence of suction on the growth of Görtler vortices decreases as
$\lambda _z^\ast$
increases, in both subsonic and supersonic conditions.
Perturbation velocity and temperature with (dash–dot) and without (solid) suction at
${\textit{Ma}}=0.80$
(a) and
${\textit{Ma}}=2.95$
(b).
The normalised velocity and temperature perturbation profiles at different streamwise locations are plotted in figure 7 for
$\gamma _w=0$
(solid curves, figure 7
a,b),
$\gamma _w=-5$
(subsonic, figure 7
a) and
$\gamma _w=-10$
(supersonic, figure 7
b). The peaks of the fluctuations move to higher
$y$
(but lower
$\eta$
) as the boundary layer thickens when suction is absent or mild. When suction is intense enough to induce the decay of the streaks, the shape of the normalised perturbation profiles is unaltered. A marked thinning of the boundary layer prevents the exponential growth of the vortices because the ratio
$\delta ^\ast /\lambda _z^\ast$
is reduced to a constant, thereby inducing a compression of the three-dimensional spanwise-adjacent structures (Fransson & Alfredsson Reference Fransson and Alfredsson2003). The thinning of the boundary layer inhibits the generation of the spanwise and wall-normal pressure gradient and the spanwise viscous diffusion of the disturbances, both instrumental in the growth of the Görtler vortices. Since
$\hat {x}=x/{\textit{Re}}=O(\delta ^{2})$
and
$\delta \to 0$
, the flow tends to be described by the boundary-layer equations, which, for solid walls, apply at locations closer to the leading edge where the perturbations grow at a slower rate (Viaro & Ricco Reference Viaro and Ricco2019a
). This slower growth is due to the
${\mathcal{Y}}$
-momentum equation (2.15c
) not entering the dynamics at leading order in the boundary-layer equations, analogous to the laminar base flow. The curvature effects that are responsible for the inviscid pressure-centrifugal imbalance are therefore negligible.
Comparison with self-similar suction: effect of
$\gamma _w$
and
$\lambda _z^\ast$
on the maximum of the streamwise-velocity disturbances in steady conditions at
${\textit{Ma}}=0.8$
(a) and
${\textit{Ma}}=2.95$
(b). The magenta and orange curves show the cases
$\lambda _z^\ast =1\,\textrm {mm}$
and
$2\,\textrm {mm}$
, respectively.
The effect of self-similar suction
$V (\hat {x},0 )=V_0/ (2\hat {x} )^{1/2}$
where
$V_0{\textit{Re}}=-5$
, is shown in figure 8 (thick curves) and compared with the results of figure 6 (thin curves) for
$\gamma _w=-5$
in the subsonic (figure 8
a) and supersonic (figure 8
b) cases. In the case of self-similar suction, the boundary-region equations (2.14) reduce to those of Viaro & Ricco (Reference Viaro and Ricco2019a
). Self-similar suction (thick curves) significantly attenuates the streaks and prevents the onset of the vortices in the subsonic case, thereby damping the disturbance growth right from its inception at the leading edge. The blowing velocity decreases downstream and self-similar suction is outperformed by the uniform suction with
$\gamma _w=-5$
and
$\lambda _z^\ast =1\,\textrm {mm}$
(magenta curves). An analogous behaviour is reported in the supersonic case (figure 3
b). The curves for the self-similar and non-similar suction cases differ in both shape and amplitude and the results cannot be reconciled by rescaling the plots.
The effect of the free stream gust frequency on the growth of Görtler vortices under the same conditions of figure 6 is shown in figure 9(a) (
${\textit{Ma}}=0.80$
,
${G}=905$
,
${\textit{Re}}=3771$
and
$x_s^\ast =4.2\,\textrm {cm}$
). A frequency of
$1260\,\textrm {Hz}$
is representative of wind-tunnel conditions at moderate Mach numbers (Viaro & Ricco Reference Viaro and Ricco2019a
; Xu et al. Reference Xu, Ricco and Duan2024). Uniform suction attenuates steady and unsteady vortices and, when
$\gamma _w=-5$
, the effect of frequency is negligible. When the boundary layer becomes thinner due to suction, the dynamics is analogous to that near the leading edge. As
$\hat x \to 0$
, the asymptotic behaviour is described by a regular perturbation of the boundary-region equations, expressed as a series summation (Leib et al. Reference Leib, Wundrow and Goldstein1999; Viaro & Ricco Reference Viaro and Ricco2019a
). The first terms in this series are governed by the steady boundary-layer equations, which explains the negligible influence of unsteadiness.
The influence of
$x_s^\ast$
on the evolution of the steady vortices is shown in figure 9(b). The growth rate for
$\gamma _w=-2$
does not depend on
$x_s^\ast$
, while the value of maximum streamwise velocity at a given
$x^\ast$
increases with
$x_s^\ast$
. For
$\gamma _w=-5$
, the disturbance growth is no longer quasiexponential and settles to an almost constant amplitude for relatively low
$x_s^\ast$
. For the largest tested
$x_s^\ast$
, the boundary layer has grown thicker and the same control is not as effective.
(a) Effect of
$\gamma _w$
on steady and unsteady Görtler vortices. (b) Effect of
$\gamma _w$
and
$x_s^\ast$
on steady Görtler vortices. For both graphs,
${\textit{Ma}}=0.8$
,
${G}=905$
and
${\textit{Re}}=3771$
. The pattern of the curves denotes the suction rate
$\gamma _w=0$
(solid) and
$\gamma _w=-5$
(dash–dot).
3.3. Neutral stability curves
The mathematical formulation presented in § 2 incorporates the receptivity of the base flow to free stream vortical disturbances, allowing a rigorous study of the evolution of the vortices and the plotting of neutral curves for the Görtler instability. The shape of the neutral curves, – i.e. curves in the parameter space that distinguish conditions of growth and decay – is significantly influenced by the initial conditions (Hall Reference Hall1983), which can only be determined by matching with the outer flow, i.e. by taking receptivity into account. As suggested by Kobayashi (Reference Kobayashi1972), the neutral curves can be easily interpreted if drawn in terms of
$G$
(2.16), which only depends on the free stream conditions and wall curvature, and not on the boundary-layer thickness. We therefore represent the curves as the points of the
$\hat {x}$
–
$G$
parameter space that satisfy the condition
$\varsigma (\hat {x} ) = 0$
, where
$\varsigma (\hat {x} ) = {\textrm{d}} E (\hat {x} )/{\textrm{d}}\hat {x}$
and
\begin{equation} E(\hat {x}) = \int _0^\infty \vert \bar u (\hat {x},\eta)\vert ^2{\textrm{d}}\eta \end{equation}
is the scaled perturbation energy divided by
$ (2\hat {x} )^{1/2}$
(Hall Reference Hall1983; Viaro & Ricco Reference Viaro and Ricco2019b
). The curves in figure 10, drawn for given
${\mathcal{F}}$
,
$\textit{Ma}$
and
$\hat {x}_s$
, separate stable (
$\varsigma \lt 0$
) and unstable (
$\varsigma \gt 0$
) regions, with colours denoting the suction rate
$\gamma _w$
. The curves embody the convective nature of Görtler instability: a perturbation evolving in
$\hat {x}$
at a fixed
$G$
grows in unstable regions and decays in stable regions. The influence of the curvature radius on the Görtler instability when all the other physical parameters are fixed is obtained by changing
$G$
at fixed
$\hat {x}$
in the maps of figure 10. When
$r_c$
is large,
$G$
is small, the perturbations evolve as laminar streaks and eventually decay by viscous effects. As
$r_c$
decreases,
$G$
increases and the laminar streaks become Görtler vortices as they undergo a quasiexponential growth.
Neutral stability maps of steady (
${\mathcal{F}}=0$
) Görtler vortices with suction for
$k_y=2$
(solid curves) and
$k_y=1$
(dashed curves) at
${\textit{Ma}}=0.8$
(a) and
${\textit{Ma}}=2.95$
(b) for the conditions of table 1 and
$\hat {x}_s=0.1$
. The maximum stable Görtler number
$G_A$
is shown as a function of
$\gamma _w$
for
$k_y=2$
in the inset in (a).
In the subsonic case (
${\textit{Ma}}=0.8$
, figure 10
a), the stable region (
$\varsigma \lt 0$
) expands considerably and the marked peak near the leading edge becomes more pronounced as
$\gamma _w$
increases. The broadening of the stable region and the reduction in the growth rate caused by increasing suction are similar to those obtained by increasing
$\textit{Ma}$
at constant
$\hat {x}$
and
${\textit{Re}}$
.
The effect of suction is much weaker in the supersonic case (
${\textit{Ma}}=2.95$
, figure 10
b), where the stable region does not broaden as significantly as in the subsonic case, especially downstream of its leading-edge peak. The neutral curves in absence of suction are broader for supersonic Mach numbers because of the stabilising effect of compressibility (Viaro & Ricco Reference Viaro and Ricco2019a
). When suction is applied, the suction term in the wall boundary condition (2.12) reduces as
$T_w$
increases. The narrow vertical band in figure 10(a) is due to the transient behaviour shown in the insets of figure 6(b,d), where suction decreases downstream of
$\hat {x}_s$
, reaches a minimum, and increases again. As
$\gamma _w$
increases, the stability region broadens upstream and eventually merges with this band (refer to yellow neutral curve in figure 10
b). The maximum stable Görtler number
${G}_{\!A}$
– shown for the peak of the curves with
$k_y=2$
– increases quasiexponentially with
$\vert \gamma _w\vert$
(inset in figure 10
a). This change of
${G\!}_A$
with
$\vert \gamma _w\vert$
is more pronounced in the subsonic case. Higher suction rates shift the location where the boundary-layer perturbations become independent of the initial and outer conditions farther downstream.
The importance of receptivity is revealed unequivocally in the influence of the wall-normal wavenumber
$k_y$
, a quantity that only enters the problem through the free stream boundary conditions. The curves for
$k_y=2$
(solid curves in figure 10) show a pronounced peak near the leading edge (letter
$A$
) that is almost absent for
$k_y=1$
(dashed). The solid and dashed curves merge at large
$\hat {x}$
and the disturbance growth is less impacted by receptivity as the perturbations evolve downstream. The reason for this result resides in the decay of the external perturbations by viscous dissipation, as evident in (2.19). The eigenvalue approach becomes tenable because the external forcing is uninfluential sufficiently downstream (Viaro & Ricco Reference Viaro and Ricco2019a
). The highest stable Görtler number
$G_A$
and its location on the neutral map
$\hat {x}_A$
are sensitive to variations in
$k_y$
. Here
$G_A$
increases with
$k_y$
when
$k_y\leqslant 6$
and decreases for
$k_y\gt 6$
(figure 11
a), while
$\hat {x}_A$
moves closer to the leading edge as
$k_y$
increases and
$k_y\leqslant 6$
, but moves slightly downstream for
$k_y\gt 6$
(figure 11
b). The increase of
$G_A$
and the decrease of
$\hat {x}_A$
observed for
$k_y\leqslant 6$
are due to the decaying forcing term in the outer boundary conditions (2.19a
) and represent the broadening of the stability region in the parameter space
$\hat {x}-G$
. The presence of moderate wall suction (red curves in figure 11) does not alter the shape of the curves, which shift to higher
$G_A$
and smaller
$\hat {x}_A$
.
Effect of the wall-normal wavelength of the external disturbance
$k_y$
on the maximum stable Görtler number
$G_A$
(a) and its location
$\hat {x}_A$
(b). The points were computed for increasing suction rates at
${\textit{Ma}} = 0.8$
.
Neutral stability maps of unsteady Görtler vortices for
$k_y=1$
and
${\textit{Ma}} = 2.95$
. The black and red curves represent the cases
$\gamma _w=0$
and
$\gamma _w=-5$
, respectively. (a) Cases
${\mathcal{F}}=10$
(solid) and
${\mathcal{F}}=80$
(dashed). (b) Case
${\mathcal{F}}=120$
. The grey curve shows the steady case without suction (
$k_y=1$
,
${\mathcal{F}}=0$
,
$\gamma _w=0$
) for comparison.
(a) Neutral maps of Görtler instability (solid curves) at moderate Mach numbers and relatively high disturbance frequency. The stable regions (
$S_1$
and
$S_2$
) are enclosed within the solid curves and the regions of weak TS growth (
${TS}_w$
) within the dashed curves. (b) Evolution of
$\vert \bar u\vert _{max}$
for
${G}=650$
(thick) and
${G}=1600$
(thin). For the suction cases, the segments of the curves describing weak (TS) growth and quasiexponential (Görtler) growth are enclosed in the dashed red rectangles. For both graphs,
${\textit{Ma}}=2.95$
,
$k_y=1$
,
${\mathcal{F}}=150.3$
and
$\gamma _w=0$
(black) and
$\gamma _w=-5$
(red).
The maps in figure 12 show the effect of wall suction on the stability of unsteady Görtler vortices. The shape of the curves is affected by unsteadiness and the stability region expands uniformly for increasing
${\mathcal{F}}$
. For low and moderate disturbance frequency (
${\mathcal{F}}=10$
, solid curves), the curves expand while retaining the same shape. For larger frequency, the stability region reaches a peak in the vicinity of the leading edge (
${\mathcal{F}}=80$
, dashed curves) and the neutral curve plateaus downstream, reaching the minimum denoted by the letter B in figure 12. The Görtler number relative to point
$B$
decreases abruptly when the disturbance frequency is increased above
${\mathcal{F}}\gt 80$
, and a narrow unstable region appears immediately downstream of the leading-edge peak (
${\mathcal{F}}=120$
, dash–dot curves).
The neutral stability curves of figure 13(a) show the effect of suction for relatively high disturbance frequency
${\mathcal{F}}$
in supersonic flows. The shape of this neutral curve for unsteady conditions is markedly different from the map of figures 10 and 12. As
${\mathcal{F}}$
increases, the peak denoted by the letter
$A$
becomes more pronounced, the minimum denoted by
$B$
drops dramatically, and the stable region eventually splits in two branches denoted by
$S_1$
and
$S_2$
in figure 13(a). The flow experiences three types of instability: laminar streaks, Görtler vortices and oblique TS waves. The latter are generated by a wavelength-shortening mechanism related to the spanwise pressure gradient of the disturbance, as first shown by Ricco & Wu (Reference Ricco and Wu2007). For moderate
$G$
, the disturbance growth in absence of suction is described by the thick black curve for
${G}=650$
in figure 13(b). Laminar streaks develop upstream of (and attenuate within)
$S_1$
. Downstream of
$S_1$
, they either decay in
$S_2$
or amplify again as TS waves, their growth being independent of the curvature (Viaro & Ricco Reference Viaro and Ricco2019b
). A strong or weak TS growth occurs depending on whether
$\beta = \textrm{d}^2\vert \bar u\vert _{max}/\textrm{d}\hat {x}^2$
is positive or negative, respectively. Regions of
$\beta \lt 0$
are enclosed in the dashed curves in figure 13(a). Immediately downstream of
$S_1$
,
$\beta$
is positive and the TS waves undergo an intense growth (
$\beta \gt 0$
). The disturbances then enter the region enclosed by the dashed curve and turn into weak TS waves (white marker
${TS}_w$
in figure 13
a). Only for very large
$G$
the perturbations turn into quasiexponentially growing Görtler vortices, as shown by the thin black curve for
${G}=1600$
in figure 13(b).
Uniform suction with
$\gamma _w=-5$
(red curves in figure 13) broadens the stable regions. The branches
$S_1$
and
$S_2$
merge and all disturbances are attenuated for
${G}\lt 150$
. Region
$S_1$
broadens almost vertically, while the influence on region
$S_2$
is milder. This finding is consistent with the results for steady conditions at moderate Mach numbers (figure 10
b). The region of the weak TS waves (delimited by the dashed curves in figure 13
a) spreads and covers a large portion of the
$\hat {x}$
–
$G$
plane that would otherwise host the more powerful exponential growth of the strong TS waves or the Görtler vortices (red marker
${TS}_w$
in figure 13
a). The thick curves for
${G}=650$
in figure 13(b) illustrate the shift from strong to weak (
$\beta \lt 0$
) TS growth. When the curvature is more marked (thin curves for
$G=1600$
), suction delays the onset of the strong TS waves and the Görtler vortices farther downstream.
(a) Splitting frequency
${\mathcal{F}}_{\textit{spl}}$
as a function of the suction rate
$\gamma _w$
and the wall-normal wavenumber
$k_y$
. (b) Splitting location at
$G=0$
as a function of
$\gamma _w$
and
$k_y$
.
With uniform suction at
$\gamma _w = -5$
, the branches
$S_1$
and
$S_2$
merge again within a narrow band just below point
$B$
in figure 12. An increase in disturbance frequency
${\mathcal{F}}$
lowers point
$B$
towards the axis
$G=0$
in the neutral map. As a critical
${\mathcal{F}}_{\textit{spl}}$
is attained, the point
$B$
reaches
$G=0$
at a streamwise location
$\hat {x}_B = \hat {x}_{\textit{spl}}$
. Conversely, increasing the suction rate
$\gamma _w$
has the opposite effect, as it causes
$S_1$
and
$S_2$
to merge and shifts point
$B$
to higher Görtler numbers. A higher
${\mathcal{F}}_{\textit{spl}}$
is required to split the stable region when suction is applied. As shown in figure 14,
${\mathcal{F}}_{\textit{spl}}$
increases linearly with the suction rate
$\gamma _w$
(figure 14
a), while the streamwise location of the splitting point
$\hat {x}_{\textit{spl}}$
decreases as
$\gamma _w$
increases (figure 14
b). For the case of the neutral map in figure 13 (
${\textit{Ma}}=2.95$
and
$k_y=1$
), the splitting frequency is
${\mathcal{F}}_{\textit{spl}}=137$
in absence of suction (black curve in figure 13
a), but increases to
${\mathcal{F}}_{\textit{spl}}=181$
when a suction rate
$\gamma _w=-5$
is applied (red curve in figure 13
a). As in the subsonic case, a moderate increase in the wall-normal wavenumber
$k_y$
has a stabilising effect on the neutral curves. The lines in figure 14(a) shift to higher splitting frequencies for
$k_y=2$
(red) and
$k_y=3$
(yellow), while the curves in figure 14(b) move downwards. This behaviour is consistent with the results reported in figures 10 and 11, which show a rapid upstream shift of the stable peak near the leading edge when
$k_y$
increases in the range
$1\leqslant k_y\leqslant 6$
.
4. Conclusions
In this study, we have investigated the effect of uniform wall suction on compressible boundary layers evolving over streamwise-concave surfaces and exposed to free stream vortical disturbances. With the long-term objective of using suction as an effective method for transition delay, we considered suction rates that are achievable across thin porous layers of small permeability and negligible surface roughness.
A laminar boundary layer developing over a suction actuator undergoes a complex evolution from the impermeable leading-edge region to its asymptotic-suction state. We have validated the results of the non-similar laminar base flow with suction against incompressible and supersonic wind-tunnel data and DNSs. To study the receptivity of the base flow to oncoming free stream vortical disturbances, we have employed the non-similar compressible boundary-region framework which has been formulated and solved numerically for the first time. The non-similarity of the viscous layer modifies the asymptotic matching between the external flow and the boundary layer, and thus the free stream boundary conditions.
Our results demonstrate that wall suction significantly reduces the amplitude of compressible Görtler vortices, expanding the region of stability in the
$\hat {x}-G$
parameter space. These findings indicate that suction can be an effective method for controlling the early stages of transition for boundary layers evolving over streamwise-concave surfaces for subsonic and moderately supersonic Mach numbers. The suppression effect weakens as the Mach number (and, therefore, the wall temperature) increases, which suggests that wall suction might not be effective in the hypersonic regime. It may serve nevertheless the purpose of delaying transition in boundary layers evolving over the nozzles of supersonic wind tunnels, thus enhancing their performance, and as a control method for boundary layers over high-speed vehicles. We have computed the neutral stability curves of compressible Görtler vortices with uniform suction, quantifying the enlargement of the stability regions as the suction rate increases.
Future work should focus on extending our formulation to take into account nonlinearity and the secondary instability of the compressible Görtler vortices as next steps to evince the influence of wall suction on transition delay. More complex free stream forcing conditions or three-dimensional base flows could be considered to obtain a more complete picture of the effect of suction, which cannot be captured by our simplified approach. Suction non-uniformity at the pore scale and the coexistence of acoustic and vortical disturbances should also be studied. The lack of experimental data of suction through concave porous walls in the compressible regime represents a critical issue to be addressed, while future experiments on transitional flows should prioritise velocity field measurements in cross-flow planes to identify where vortical structures first emerge within the boundary layer.
Acknowledgements
L.F. is grateful to Dr D. Xu, Dr K. Zhu and Dr M. Falcone for their comments and support. L.F. and P.R. thank the reviewers for their insightful comments that improved the quality of the article.
Funding
The authors acknowledge the financial support of the US Air Force through AFOSR grant FA8655-21-1-7008 (International Program Office – Dr D. Smith). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the funding agencies or the US Government. L.F. was also supported by the University of Sheffield’s Postgraduate Research Student Publication Scholarship 2024.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Outer behaviour of the base-flow wall-normal velocity
The velocity components of the steady, two-dimensional outer flow in region IV
$\lbrace 1,0,0\rbrace + \epsilon \lbrace \acute u, \acute v, 0 \rbrace (\hat {x},\hat {y})$
(2.20) are described by the stream function
$\varPsi (\hat {x},y,\hat {y} ) = y + \psi _1 (\hat {x},\hat {y} )$
:
\begin{align} U_{\textit{out}} &= 1 + \epsilon (\acute \rho + \acute u) = \frac {\partial \varPsi }{\partial y} = 1 + \frac {1}{{\textit{Re}}}\frac {\partial \psi _1}{\partial \hat {y}}, \end{align}
\begin{align} V_{\textit{out}} &= \epsilon \acute v = - \frac {1}{{\textit{Re}}}\frac {\partial \varPsi }{\partial \hat {x}} = - \frac {1}{{\textit{Re}}}\frac {\partial \psi _1}{\partial \hat {x}}. \end{align}
Substitution of (2.20) in the Navier–Stokes equations and the perfect gas equation yields the Euler equations for
\begin{align} \frac {\partial \acute u}{\partial \hat {x}} + \frac {\partial \acute p}{\partial \hat {x}} &= 0, \end{align}
\begin{align} \frac {\partial \acute v}{\partial \hat {x}} + \frac {\partial \acute p}{\partial \hat {y}} &= 0, \end{align}
\begin{align} \frac {\partial \acute \tau }{\partial \hat {x}} - \left (\gamma - 1\right ){\textit{Ma}}^2\frac {\partial \acute p}{\partial \hat {x}} &= 0, \end{align}
\begin{align} \gamma {\textit{Ma}}^2\acute p - \acute \rho - \acute \tau &= 0. \end{align}
By assuming that shock waves are absent, one can write (A2c ) as
\begin{equation} \frac {\partial \acute \tau }{\partial \hat {y}} - \left (\gamma - 1\right ){\textit{Ma}}^2\frac {\partial \acute p}{\partial \hat {y}} = 0, \end{equation}
and combining (A3), (A2b ), (A2d ) and (A1b ) yields
\begin{equation} {\textit{Ma}}^2\frac {\partial ^2\psi }{\partial \hat {x}^2} + r_t\left (\frac {\partial \acute u}{\partial \hat {y}} - \frac {\partial \acute v}{\partial \hat {x}}\right ) = \frac {\partial ^2\psi _1}{\partial \hat {y}^2} + \frac {\partial ^2\psi }{\partial \hat {x}^2}. \end{equation}
Because the inviscid outer flow is irrotational, the stream function
$\psi _1$
is governed by the Poisson equation
\begin{equation} (1 - {\textit{Ma}}^2)\frac {\partial ^2\psi _1}{\partial \hat {x}^2} + \frac {\partial ^2\psi _1}{\partial \hat {y}^2} = 0. \end{equation}
Equation (A5) is solved by matching (A1b
) with the large-
$\eta$
limit of the wall-normal velocity component (2.11b
) given in (2.21). The wall-normal velocity
$V_{\textit{out}}$
decays in the far field for
$\hat {y}\gg 1$
. The boundary condition at the plate is recovered by matching the inner and outer limits of (A1b
) and (2.11b
), respectively. The stream function
$\psi _1$
at
$\hat {y}\ll 1$
and
$y=O(1)$
is given by (2.23).
In self-similar, compressible boundary layers (2.23) reduces to
\begin{equation} \psi _1 (\hat {x},0) = - (2\hat {x})^{1/2} \left (\gamma _c + \beta _c\right )\! . \end{equation}
If the flow is subsonic, (A5) is elliptic and can be solved analytically using complex-variable theory (van Dyke Reference van Dyke1975, p. 136)
\begin{equation} \varPsi \left (x,y\right ) = y - \frac {1}{{\textit{Re}}^{1/2}}{{\rm Re}}\left \lbrace \left [2x + 2{\textrm{i}}(1 - {\textit{Ma}}^2)^{1/2}y\right ]^{1/2}\left (\beta _c + \gamma _c\right )\right \rbrace \!, \end{equation}
where
$\textrm {Re}$
denotes the real part. Equation (A7) reduces to (5.8) of LWG99 in the incompressible case (
${\textit{Ma}}=0$
and
$\gamma _c=0$
).
Appendix B. Numerical solution of the compressible boundary-region equations on non-uniform grids
A schematic of the streamwise and wall-normal grid is shown in figure 15(a). A Keller-box method is used to compute the base flow. The method is a second-order block-elimination algorithm that computes (2.5) on the staggered grid centred on the red point
$(\hat {x}_{n-1/2},\eta _{j-1/2})$
and uses the variables defined at the vertices of the red rectangle
$(\hat {x}_n,\eta _j)$
,
$(\hat {x}_n,\eta _{j-1})$
,
$(\hat {x}_{n-1},\eta _{j-1})$
and
$(\hat {x}_{n-1},\eta _j)$
. For further details, the reader is referred to Cebeci (Reference Cebeci2002).
(a) Schematic of the non-uniform spatial discretization used in the computation of the boundary-region equations. (b) Base-flow streamwise velocity profile
$U=F^\prime$
in the near wall region. A uniform (magenta triangles) and a Chebyshev–Gauss–Lobatto (CGL) grid (black crosses) were used to collocate the same number of points
$N_p=3000$
on a width
$\eta _{max}=30$
.
Once
$F$
and
$T$
are known, a finite difference scheme is employed to compute the velocity components,
$\bar u$
,
$\bar v$
,
$\bar w$
, the temperature
$\bar \tau$
and the pressure
$\bar p$
. The terms that multiply
$\bar u$
,
$\bar v$
,
$\bar w$
,
$\bar \tau$
and
$\bar p$
are defined at the point
$(\hat {x}_n,\eta _j)$
denoted by the olive square. A uniform marching step is used in
$\hat {x}$
, the streamwise derivatives are estimated by backward differencing and the variables at
$(\hat {x}_{n-1},\eta _j)$
and
$(\hat {x}_{n-2},\eta _j)$
are stored from upstream computations. The wall-normal derivatives are approximated with finite differences schemes centred in
$(\hat {x}_n,\eta _j)$
:
\begin{align} \left (\frac {\partial \bar u}{\partial \eta }\right )_{\!j} &= \frac {{\mathcal{C}}_j\bar u_{j+1} + (1-{\mathcal{C}}_j)\bar u_j - \bar u_{j-1}}{{\mathcal{C}}_j\Delta \eta _{j+1} + \Delta \eta _j}, \end{align}
\begin{align} \left (\frac {\partial ^2 \bar u}{\partial \eta ^2}\right )_{\!j} &= 2\frac {\mathcal{P}_j\bar u_{j+1} - (1+\mathcal{P}_j)\bar u_j + \bar u_{j-1}}{\mathcal{P}_j\Delta \eta _{j+1}^2 + \Delta \eta _j^2}, \\[9pt] \nonumber \end{align}
where the weights are
${\mathcal{C}}_j = (\Delta \eta _j/\Delta \eta _{j+1})^2$
and
$\mathcal{P}_j = \Delta \eta _j/\Delta \eta _{j+1}$
. Upward finite differences are used to enforce the adiabatic condition at the wall
$j=1$
:
\begin{equation} \left (\frac {\partial \bar \tau }{\partial \eta }\right )_1 = \frac {{\mathcal{C}}_{\textit{up},1}\bar \tau _{3} - \bar \tau _{2} + (1-{\mathcal{C}}_{\textit{up},1})\bar \tau _1}{{\mathcal{C}}_{\textit{up},1}\left (\Delta \eta _{3} + \Delta \eta _2\right ) - \Delta \eta _2}=0, \end{equation}
and
${\mathcal{C}}_\textit{up},1=\Delta \eta _2^2/ (\Delta \eta _{3} + \Delta \eta _2 )^2$
. Downward finite difference schemes are used to enforce the boundary conditions of the mixed type (2.19) at the outer boundary
$j=N_p$
– where
$N_p$
is the number of points in the wall-normal direction –
\begin{equation} \left (\frac {\partial \bar v}{\partial \eta }\right )_{N_{p}} = \frac {{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}}\bar v_{N_{p}-2} - \bar v_{N_{p}-1} - ({\mathcal{C}}_{\textit{do}w\textit{n},N_{p}}-1)\bar v_{N_{p}}}{\Delta \eta _{N_{p}} - {\mathcal{C}}_{\textit{do}w\textit{n},N_{p}}(\Delta \eta _{N_{p}} + \Delta \eta _{N_{p}-1}}, \end{equation}
where
${\mathcal{C}}_{\textit{do}w\textit{n},N_{p}}=\Delta \eta _{N_{p}}^2/(\Delta \eta _{N_{p}} + \Delta \eta _{N_{p}-1})^2$
. The asymptotic outer boundary conditions are imposed by introducing
\begin{align} \bar v_{N_{p}} & = \frac {{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}} \bar v_{N_{p}-2} - \bar v_{N_{p}-1} - \mathcal{A}_{N_{p}}{\textrm{e}}^{{\textrm{i}} {\mathcal{F}}\hat {x} + {\textrm{i}} k_y\left (2\hat {x}\right )^{1/2}\left [\eta -\beta _c\left (\hat {x}\right )\right ]- \big(k_y^2+k_z^2\big)\hat {x}}}{{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}} - 1 + \mathcal{A}_{N_{p}}\left \vert k_z\right \vert (2\hat {x})^{1/2}}, \end{align}
\begin{align} \bar w_{N_{p}} &= \frac {{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}} \bar w_{N_{p}-2} - \bar w_{N_{p}-1}}{{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}} - 1 + \mathcal{A}_{N_{p}}\left \vert k_z\right \vert (2\hat {x})^{1/2}} + \nonumber \\ &\quad + \frac {\mathcal{A}_{N_{p}}{\textrm{i}} k_y (2\hat {x})^{1/2}{\textrm{e}}^{{\textrm{i}} {\mathcal{F}}\hat {x} + {\textrm{i}} k_y\left (2\hat {x}\right )^{1/2}\left [\eta -\beta _c\left (\hat {x}\right )\right ]- \big(k_y^2+k_z^2\big)\hat {x}}}{{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}} - 1 + \mathcal{A}_{N_{p}}\left \vert k_z\right \vert (2\hat {x})^{1/2}}, \end{align}
\begin{align} \bar p_{N_{p}} &= \frac {{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}} \bar p_{N_{p}-2} - \bar p_{N_{p}-1}}{{\mathcal{C}}_{\textit{do}w\textit{n},N_{p}} - 1 + \mathcal{A}_{N_{p}}\left \vert k_z\right \vert (2\hat {x})^{1/2}}, \end{align}
- where
$\mathcal{A}_{N_{p}}={\mathcal{C}}_{\textit{do}w\textit{n},N_{p}}(\Delta \eta _{N_{p}} + \Delta \eta _{N_{p}-1}) - \Delta \eta _{N_{p}}$
- in the discretization at
$\eta _{N_{p}-1}$
. The pressure is computed on a staggered grid (blue circles in figure 15) to avoid the odd–even decoupling
\begin{equation} \bar p_{j+1/2} = \frac {\bar p_{j+1} + \bar p_j}{2}, \end{equation}
and the wall-normal pressure gradient is
\begin{equation} \frac {\partial \bar p}{\partial \eta } = \frac {\bar p_{j+1}-\bar p_j}{\Delta \eta _{j+1}}. \end{equation}
A CGL grid is used to distribute the grid points (olive points in figure 15 a) in the wall-normal direction (Aref & Balachandar Reference Aref and Balachandar2018). The location of each point is given by
\begin{equation} \eta _j = \eta _{max} \left [1-\cos \left (\frac {\pi j}{2 N_{p}}\right )\right ]\mbox{ for }0\leqslant j\leqslant N_p, \end{equation}
in the range
$\eta \in [0,\eta _{max} ]$
. The base-flow results obtained on uniform and CGL grids with
$N_p=3000$
and
$\eta _{max}=30$
are compared in figure 15(b). The CGL grid (black crosses) is substantially more refined than the uniform grid (magenta triangles) in the near-wall region. Grid independence in
$\eta$
was verified by increasing
$\eta _{max }$
to 60 and
$N_p$
to 6000. Grid independence in
$\hat {x}$
was ensured by repeating the computations with different marching steps:
$\Delta \hat {x} = 10^{-3}$
,
$5 \times 10^{-4}$
,
$10^{-4}$
and
$5 \times 10^{-5}$
. We adopted
$\Delta \hat {x} = 10^{-4}$
for the computations presented in this paper.
Validation of the disturbance flow without suction computed on the CGL grid. (a) The growth of the steady, incompressible Görtler vortices compared with the experimental data of Tani (Reference Tani1962, T62) and Finnis & Brown (Reference Finnis and Brown1997, FB97) as in Wu et al. (Reference Wu, Zhao and Luo2011). (b) The growth of compressible steady Görtler vortices compares well with the results of (Viaro & Ricco Reference Viaro and Ricco2019a
, figure 9a) for
${\textit{Ma}}=4.0$
and
$k_y=1$
.
To validate the algorithm and the use of the CGL grid, we followed Wu et al. (Reference Wu, Zhao and Luo2011), and compared our solution with (2.15a
)–(2.15e
) – which reduce to the form of Viaro & Ricco (Reference Viaro and Ricco2019a
) in absence of suction – against the incompressible wind tunnel data of Tani (Reference Tani1962, T62) and Finnis & Brown (Reference Finnis and Brown1997, FB97). Since the amplitude of the free stream gusts
$\epsilon$
was not provided in T62 or in FB97, the numerical results were rescaled with respect to the experimental data point located farthest upstream (Xu et al. Reference Xu, Zhang and Wu2017). The agreement is good for all the cases (refer to figure 16). Unfortunately, a direct comparison with the supersonic wind tunnel data of Wang et al. (Reference Wang, Wang and Zhao2018) is not possible and an experimental validation cannot be performed in the compressible regime. As shown in figure 16(b), our code reproduces the computations of Viaro & Ricco (Reference Viaro and Ricco2019a
) (figure 9a therein) for the case
${\textit{Ma}}=4.0$
and
$k_y=1$
and different Görtler numbers.
Finally, the discrete points of the neutral stability curves in § 3.3 were computed by solving the CLUBR equations for increasing
$G$
and storing the streamwise location of the neutral point. In most cases, the neutral curves are functions
${G}_{\varsigma } (\hat {x} )$
, where
${G}_{\varsigma }$
is the Görtler number corresponding to a neutral point, and a Quicksort algorithm (Hoare Reference Hoare1962) was used to sort the detected points for ascending
$\hat {x}$
.
