Buoyancy-driven algebraic (localised) boundary-layer disturbances

Algebraically developing (two-dimensional) disturbance modes in a flat-plate boundary layer were described in the early work of Libby and Fox [1] and Stewartson [2]. For a dimensionless downstream coordinate x, with leading edge at \(x=0\) and transverse coordinate y these two-dimensional flows are described by the stream functionwhere \(\epsilon \ll 1\) is the disturbance amplitude, Y is the boundary-layer coordinate \(Y=y\; \text {Re}^{1/2}\) and \(\text {Re}\gg 1\) a Reynolds number. Here F satisfies the Blasius ordinary-differential equationA one-dimensional eigenvalue problem is obtained for the disturbance shape \(f({\eta })\) and eigenvalue \(\lambda \). Libby and Fox showed that such modes decay downstream with \(\lambda \le -1\); Stewartson had already showed that a disturbance with \(\lambda =-1\) can be found explicitly, which turns out to be the slowest decaying two-dimensional disturbance for the Blasius boundary layer. Although the focus of the Libby and Fox analysis was on providing a perturbation approximation to the downstream solution, it should be noted that the self-similar form of (1) applies globally at all downstream locations.

The two-dimensional Blasius problem was subsequently extended to a mixed-convection boundary layer, both with and without an externally applied pressure gradient. This extension proved to be non-trivial and has some peculiar properties that were the root cause of confusion in the early literature, particularly for the boundary-layer flow above a cold surface as described by Schneider and Wasel [3], Wickern [4, 5] and Daniels [6]. These inconsistent results were ultimately clarified by the local analysis of Steinrück [7], who showed that two-dimensional disturbances grow downstream with a growth rate that becomes unbounded. This susceptibility to what may be thought of as short-scale perturbations causes significant issues to numerical procedures for downstream marching of the parabolic governing equations for two-dimensional mixed-convection boundary layers.

The Steinrück analysis took a local parallel flow approach to describe the disturbance field, which can be justified in the limit of unbounded downstream spatial growth. However, a more general approach was later presented by Denier et al. [8] by an examination of algebraic modes of the form (1) extended to include a temperature field and assuming a Grashof number of \(O(\text {Re}^{5/2})\) in terms of a Reynolds number \(\text {Re}\). For positive Grashof numbers (a heated surface), the eigenvalue problem was solved to show that \(\lambda <0\) (decaying relative to the base flow) in the absence of pressure gradients. In the forced-convection limit the Libby and Fox [1] eigenvalues of (1) still apply, but there is now a slower-decaying (\(\lambda \approx -0.296\) for a Prandtl number \(\text{ Pr } \approx 0.7\)) disturbance associated primarily with the energy equation. For negative Grashof numbers (a cooled surface), growing \(\lambda >0\) disturbances are recovered which, in agreement with the observations of Steinrück [7], become unbounded (\(\lambda \rightarrow \infty \)) as the scaled Grashof number approaches zero from below (with no external pressure gradient). These infinite growth-rate ‘short-scale’ waves owe their presence to the large Grashof number (\(\text {Gr}\sim \text {Re}^{5/2}\)) and the resultant coupling with a streamwise pressure gradient. They effectively render the parabolic two-dimensional mixed-convection problem ill-posed. Denier et al. demonstrated that short-scale perturbations can be suppressed by specifying Neumann conditions at the downstream boundary coupled with a quasi-elliptic approach to the computations. However, for three-dimensional disturbances (as we shall discuss here) a buoyancy coupling is achieved with a weaker thermal forcing (\(\text {Gr} \sim \text {Re}^{3/2}\)), and this recovers a well-posed parabolic problem.

Additional extensions of (1) to capture three-dimensional disturbances were introduced by Luchini [9] by assuming spanwise periodicity for the isothermal Blasius boundary layer whilst similar descriptions were used by Duck et al. [10] in the context of isothermal corner boundary layers. The corresponding analysis presented by Luchini is inevitably local in nature, owing to the transverse length scale of the flow (the boundary-layer thickness) typically growing downstream in contrast to any assumed fixed spanwise wavelength of the disturbance field. However, as recently shown by Hall [11] the Luchini modes dominate at the leading edge, where the spanwise wavelength is large (relative to the transverse scale). These disturbances then describe a locally self-similar near wall response that grows as \(x^{0.223}\), with an outer non-self-similar layer that acts to reduce the transverse flow to zero, before developing directly into unstable Görtler vortices.

An alternative global approach to three-dimensional disturbances has recently been formulated by Hewitt and Duck [12], by assuming a spanwise length scale that remains commensurate with the transverse scale, both developing in tandem downstream [13]. In this new approach, there is spanwise diffusion and the disturbance ultimately decays in the far-spanwise direction, remaining localised (rather than periodic). It is possible to construct linearised self-similar disturbances relevant to all downstream locations by extending the form of (1) to give a velocity field ofwhere (u, v, w) are the streamwise, transverse and spanwise velocity components with \((y,z)=(Y,Z)\text {Re}^{-1/2}\) and Z is the spanwise coordinate. In this approach, localised three-dimensional modes are found to decay relative to the base flow with \(\lambda \approx -0.787\) when F is the corresponding Blasius solution of (2).

In this work, we return to the downstream development of three-dimensional disturbances in the form of (3) applied to a horizontal boundary layer, but include thermal effects leading to a buoyancy coupling with the transverse momentum. In addition, we extend the base flow from Blasius to the Falkner–Skan [14] boundary layer, where the inviscid (outer) flow is assumed to have a dimensional streamwise velocity \(U^*_{\infty } x^n\), where \(n>0\) is associated with a favourable external pressure gradient.

Away from any bounding surface we consider an incompressible fluid of known reference temperature \(T^*_\infty \) and density \(\rho ^*_{\infty }\), but allow for temperature variations of magnitude \(\varDelta T^*\) within the boundary layer. For a reference length scale of \(L^*\), we consider the high Reynolds number regime, withfor a constant kinematic viscosity \(\nu ^*\). The appropriate perturbation expansions for the dimensionless coordinates, velocities, temperature and pressure (respectively) are where \(u_e(x)=x^n\). Under a Boussinesq approximation with gravity in the \(-Y\) direction, the leading-order (steady) equations take the form subject to \(\hat{U}=\hat{V}=\hat{W}=0\) and \({\hat{\vartheta }}\) prescribed on the surface (\(Y=0\)), with \(\hat{U}\rightarrow u_e(x)\), \({\hat{\vartheta }} \rightarrow 0\) and \(\hat{W}\rightarrow 0\) as \(Y\rightarrow \infty \). Here, we have assumed an \(O(\text {Re}^{3/2})\) Grashof number and an O(1) Prandtl number, such thatwith constant specific heat \(c_p^*\), thermal conductivity \(\kappa ^*\), thermal expansion coefficient \(\alpha ^*\) and gravitational acceleration \(g^*\).

In the absence of any localised thermal forcing or instability, a self-similar two-dimensional (\({\hat{W}} \equiv 0\)) solution can be found to (6) in Falkner–Skan [14] form as follows: Here, \(F({\eta })\) and \(H({\eta })\) are determined from the Falkner–Skan equation and coupled energy equation parameterised in terms of the Hartree [15] parameter \(\beta = 2n/(n+1)\). Here, \(H_w\) is a dimensionless boundary temperature, which by choice of \(\varDelta T^*\) can be set to unity; however, here we will later consider the case of \(H_w=0\) (so \(H\equiv 0\) and there is only a spatially localised temperature difference).

The downstream (\(x^{3(n-1)/2}\)) form of the temperature is chosen to retain the buoyancy coupling in the three-dimensional problem, but this is only appropriate for \(\text {G}=O(1)\). If \(\text {G}=O(\text {Re})\) then there is additional coupling between the temperature field and the streamwise momentum equation via the downstream pressure gradient, in which case we would require surface temperature differences proportional to \(x^{(5n-1)/2}\) instead, as considered by Denier et al. [8] amongst others. Throughout this discussion we assume that \(\text {G}/\text {Re}=o(1)\), which avoids any growth of short-scale perturbations of the form described by [8].

If we instead choose a uniform heated plate, we recover the non-self-similar formulation of Hall and Morris [16] who showed that linear disturbances typically decay from an initiation site, before starting to grow further downstream as the flow becomes unstable to longitudinal vortex modes. The description of [16] is the most generic way for vortex modes to be generated in the presence of surface heating over spanwise scales that are large compared to the boundary-layer thickness.

Here, we extend the decomposition (3) (and therefore (1)) to include a temperature field and bouyancy coupling in the context of (6). Section 2 solves the resulting bi-global (linear) eigenvalue problem to obtain a disturbance field that grows algebraically like \(x^\lambda \) with \(\lambda >0\) (real). In Sect. 2.1, we show that the new eigenvalues are determined by a one-dimensional eigenvalue problem in the limit of the exterior fluid and boundary surface having the same temperature (\(\text {G}\ll 1\)). If the externally imposed pressure gradient is sufficiently large and favourable (\(\beta >\beta _c\)) then \(\lambda \) becomes negative and the modes decay downstream. In Sects. 2.2.1 and 2.2.2, we show that values for \(\beta _c\) can be obtained asymptotically in the limits of large/small Prandtl number.

Clearly, the constraint of a surface temperature variation of the form \(x^{(3n-1)/2}\) is restrictive, so we will place most attention on the case \(H_w=0\) (hence \(H\equiv 0\)). In this scenario, the bulk surface and far-field fluid are at the same temperature but we can still allow for spatially localised temperature deviations of size \(\varDelta T^*\). In this approach, the Grashof parameter \(\text {G}\) becomes a measure of the amplitude of this localised thermal forcing. In Sect. 3, we march the nonlinear and non-self-similar system (6) in x, solving in the Y, Z-plane (equivalently the \({\eta },{\zeta }\)-plane) for a localised ‘hot spot’ forcing on the boundary. We show that the growing eigenmodes are recovered downstream in the linear regime, developing into fully nonlinear (non-self-similar) streaks at larger forcing. The nonlinear downstream development provides a structure upon which inviscid secondary instabilities can grow. In Sect. 4, we briefly explore the stability to Rayleigh instabilities in the cross-section plane of a fixed downstream position.

We seek spatially developing steady eigenmodes analogous to those found for the isothermal Blasius (\(n=0\)) boundary layer by Hewitt and Duck [12] but extended to include buoyancy coupling with the energy equation and an external pressure gradient (\(n>0\), as formulated in [17]). These exist in the form withwhere the base flow is a (two-dimensional) Falkner–Skan boundary layer with \(U_B({\eta })\equiv F'({\eta })\), \(V_B({\eta })\equiv [(1-n)/(1+n)] {\eta } F'({\eta }) - F({\eta })\) and \(\vartheta _B({\eta }) \equiv H({\eta })\) as described by (8). Here \(\epsilon \ll 1\) for linear disturbances and the spanwise coordinate \({\zeta }\) is scaled with the local boundary-layer thickness such that both coordinates in the cross section are The \({\eta }\) coordinate simply matches the Falkner–Skan similarity form (8), so as the solution develops downstream the spanwise length scale remains comparable to the local boundary-layer thickness. The downstream dependence of the \(O(\epsilon )\) perturbation is entirely captured by the algebraic dependence of \(x^\lambda \) and (9d), so these modes apply at all downstream positions despite the non-parallel nature of the problem.

The dominant modes are such that \(\lambda \in \Re \) and therefore when \(\lambda > 0\) these three-dimensional eigenmodes grow algebraically downstream relative to the two-dimensional base flow velocity field. For isothermal (\({{\tilde{\vartheta }}}\equiv 0\)) disturbances applied to a Blasius boundary layer (\(n=0\)) it is known that such modes exist but that they decay (\(\lambda < 0 \)) away from the leading edge [12].

Substitution of (9) into (6) gives a bi-global eigenvalue problem for the spatial growth ‘rate’ \(\lambda \) as a function of the Prandtl number (\(\text {Pr}\)) and pressure gradient parameter (\(\beta \) or equivalently n). To avoid explicit treatment of the pressure in (6) it is convenient to cross-differentiate to eliminate \({\hat{P}}\) in favour of a vorticity component (\({{\tilde{\varTheta }}}\)) and then use a substitution which mirrors that appear in the standard Falkner–Skan solution (8). The resulting bi-global eigenvalue problem is, on replacing \(n=\beta /(2-\beta )\): where \(\bar{\nabla }^2\) is the two-dimensional Laplacian in the \(({\eta },{\zeta })\) plane. The velocity-field conditions on the plate (\({\eta }=0\)) are no-slip and impermeability, whilst the disturbance field decays to zero for sufficiently large values of \({\eta }\) or \({\zeta }\). The temperature disturbance remains zero both at the boundary and far field. This bi-global eigenvalue spectrum for \(\lambda \) will obviously include the isothermal Hewitt and Duck [12] decaying (\(\lambda <0\)) eigenmodes for \(\text {G}\ll 1\) (with \({\tilde{\vartheta }} =0\)), but is here also augmented by new thermal modes (\({\tilde{\vartheta }} \ne 0\)) which are the focus of our current discussion.

In the limit of a small Grashof parameter \(\text {G}\ll 1\) the energy equation decouples in (11); this version of the eigenproblem is also recovered if the bulk surface temperature is the same as the far-field fluid, that is \(H_w=0\) (\(H\equiv 0\)).

The problem is solved with a second-order finite-difference scheme that spans \(N_{\eta } \times N_{\zeta }\) nodes, resulting in a generalised eigenvalue problem of size \(5N_{{\eta }}N_{{\zeta }}\times 5N_{{\eta }}N_{{\zeta }}\). A non-uniform mesh is applied to improve resolution near to the origin and SLEPc (Hernandez et al. [18]) is employed to make use of (iterative) sparse matrix methods. Solutions presented below are independent of the mesh resolution and domain truncation, but typical resolutions are \(N_{\eta } = N_{\zeta } = 600\) over a truncated domain size of \(({\eta },{\zeta }) \in [0,40]\times [0,40]\).

The eigenvalue spectrum varies with the Prandtl number \(\text {Pr}\), pressure gradient \(\beta \) and Grashof parameter \(\text {G}\). As a representative example, we present eigenvalues determined from the bi-global eigenvalue problem for Prandtl number \(\text {Pr}=0.71\) (air at room temperature), no pressure gradient \(\beta =0\) and varying G in Table 1. The decaying \({\tilde{\vartheta }}\equiv 0\) modes of Hewitt and Duck [12] are present in the bi-global eigenvalue calculations with \(\text {G}\ll 1\).

The least damped eigenvalue \(\lambda _1=0.203\) presented in Table 1 for \(\text {G}\ll 1\) represents an algebraically growing thermal mode embedded in a Blasius base flow. The velocity field grows downstream with spatial growth \(x^{0.203}\) (for the streamwise velocity), whilst the temperature field decays with \(x^{-1.297}\). The first few least damped eigenmodes are only weakly varying with \(\text {G}\) (which motivates the analysis of Sect. 2.1 below) but the spatial growth of the first mode increases at sufficiently large values of \(\text {G}\).

The first three \(\text {G}= 1\) thermal eigenmodes associated with \(\lambda _1\), \(\lambda _2\) and \(\lambda _4\) of Table 1 are shown in Fig. 1. The dominant mode is shown in the top sub-figure (a). The eigenmodes are localised, and a streamwise-aligned streak is exhibited with associated in-plane rolls. The modes are assumed to be symmetric about the \({\zeta }=0\) midplane for \({\tilde{U}}, {\tilde{V}}, {\tilde{\vartheta }}\) and antisymmetric for \({\tilde{W}}\); the eigenmodes are therefore only shown for \({\zeta }>0\).

At finite values of the Grashof parameter \(\text {G}\), we solve the bi-global eigenvalue problem as described above. However, in the limit of \(\text {G}\ll 1\) it is possible to make some progress with an alternative description that provides some insight into the new (thermal) eigenvalue distribution in Table 1.

Having established the existence of growing linear eigenmodes, we now tackle the fully nonlinear and non-self-similar problem defined by (6). A reformulation of (6) in terms of a three-dimensional perturbation to a two-dimensional base flow of (8) is given in Appendix A and it is this form of the problem that we solve numerically in this section.

We restrict attention to the case of only a localised temperature difference between the surface and exterior fluid, this corresponds to taking \(H_w = 0\) in (8) such that \(H\equiv 0\). To initiate a steady disturbance an ad hoc smooth localised surface temperature is forced in the formThis forcing is centred on \(x_0\), and its streamwise and spanwise extents are set by \(\gamma \) and c. Herein, we present solutions for \(\gamma =10\) and \(c=1\) as representative values. Even assuming \(H\equiv 0\) we retain the Grashof parameter \(\text {G}\) by defining the temperature measure \(\varDelta T^*\) as the peak difference associated with the hot spot defined by (34); hence in this case the amplitude of the spatially localised forcing becomes parameterised by the Grashof parameter \(\text {G}\).

We employ parabolic marching in the streamwise x coordinate, using a second-order Crank–Nicolson finite-difference scheme for successive x locations, solving in the cross-sectional Y, Z plane. Rather than Y and Z we use the \({\eta }\) and \({\zeta }\) coordinates for computational convenience, but the scheme does not assume any self-similarity in the flow. In the cross-sectional plane the system is discretised over a non-uniform mesh of size \(N_{\eta } \times N_{\zeta }\). We start the marching procedure with trivial initial conditions at \(x=0\), at which point the forcing (34) is exponentially small and negligible (being much less than the convergence tolerance employed in the numerical scheme). At each x step thereafter and with each Newton iteration, we solve a sparse linear system of size \(5N_{\eta } N_{\zeta }\times 5N_{\eta } N_{\zeta }\). The linear solve phase is enabled by use of a sparse parallel formulation in the C++ library PETSc (Balay et al., [20]).

To assess the downstream development of the disturbances, we employ the global measure suggested by Andersson et al. [21],for which \(U_B=F'({\eta })\) is the Falkner–Skan solution and U is as defined by (9a). The measure \(\mathcal {A}\) indicates the amount that the disturbance deviates from the underlying Falkner–Skan boundary layer.

We exhibit the measure \(\mathcal {A}\) in Fig. 5 for streaks determined using a Grashof parameter of \(\text {G}=\frac{1}{2},1,2,4,8,16\) with \(\text {Pr}=0.71\). The streaks are embedded in a Blasius boundary layer (\(\beta =0\)), and the plate heating (34) is centred on \(x_0=10\). The \(x^{0.203}\) algebraic growth of the dominant linearised eigenmode given in Table 1 is also plotted as a line segment. The streaks exhibit this algebraic growth for small Grashof parameter (i.e. low forcing amplitude). Cross sections of the \(G=16\) streak depicted in Fig. 5 are shown in Fig. 6 at \(x=20\) and \(x=250\).

In the developing streak there is a downstream velocity \({\hat{U}}(x,Y,Z)\) as defined in (5). To assess the inviscid stability of this base flow, we seek a linear disturbance field of the formAs usual, this disturbance is on a short length scale (X) which is comparable to the local boundary-layer thickness, and a corresponding fast timescale T such that \((x,t)=\text {Re}^{-\frac{1}{2}}(X,T)\). At leading order this leads to the following inviscid disturbance equations: The temperature profile remains decoupled resulting in a local Rayleigh problem, thermal effects would only remain coupled if \(\text {G}\gg 1\) as described by Hall [22]. By eliminating in favour of the disturbance pressure, and by considering a local wave number \(\bar{\alpha }\) and frequency \(\bar{\omega }\) ofwe reduce (37) toTo obtain this form we have moved to the coordinates \(({\eta },{\zeta })\) (9d) that describe the streak base state, where \({\hat{U}}(x,Y,Z) = x^n U(x,{\eta },{\zeta })\). This is the Rayleigh pressure equation, see for example Hall and Horseman [23] or Hewitt and Duck [24], and spatially unstable modes have \(\bar{\alpha }_i\ne 0\). In the \(({\eta },{\zeta })\) far-field the disturbance pressure decays to \(\tilde{p}=0\), and on the plate (\({\eta }=0\)) we have \(\tilde{p}_{\eta }=0\) by (37c). Varicose modes have symmetric (about \({\zeta }=0\)) \(\tilde{p}\) whilst sinuous modes have antisymmetric \(\tilde{p}\).

To solve the bi-global polynomial eigenvalue problem (39), we employ a second-order finite difference scheme for \(\tilde{p}\), including \(\bar{\alpha }\) as an extra degree of freedom. To compensate for the extra degree of freedom we must also apply an extra constraint on the system: we arbitrarily normalise the eigenfunctions so that \(\tilde{p}=1\) at a fixed node in the domain. This approach can only determine one eigenmode at a time, but avoids working with the full polynomial (in \(\bar{\alpha }\)) nature of (39). Initial guesses to the bi-global inviscid modes are taken from the corresponding viscous problem.

In Fig. 5, the \(\text {G}=8,16\) solutions each become unstable to a Rayleigh mode downstream, with the G = 16 streak immediately exhibiting a sinuous instability at the heated region. We indicate approximate locations at which further instabilities are first exhibited: a sinuous mode for \(\text {G}=8\) (plus-sign); and a varicose mode for \(\text {G}=16\) (circle). Sinuous modes are found at lower values of the metric \(\mathcal {A}\) than varicose modes.

The downstream evolution of the peak inviscid growth rate is shown in Fig. 7 using \(\underset{\bar{\omega }}{\max }[-\bar{\alpha }_i]\), the spatial growth of the most unstable mode. The unstable points in Fig. 5 correspond to the lowest streamwise location of the unstable modes given in Fig. 7. The \(G=16\) streak is seen to be already unstable to a sinuous mode at the heated region.

The increased forcing associated with \(\text {G}=16\) results in strongly nonlinear streak-like behaviour in the immediate vicinity of the heated region (\(x=x_0\)), where unstable sinuous modes are found. The sinuous mode of the \(\text {G}=16\) streak is consistently more unstable than the varicose mode. For weaker forcing with \(\text {G}=8\), growth ultimately results in a nonlinear streak further downstream that is also unstable to an inviscid mode.

We depict the unstable sinuous and varicose modes of the \(\text {G}=16\) streak in Fig. 8 at ad hoc choices of \(x=250\) and frequency \(\bar{\omega }=0.25\). Also shown is the upwards deflection of the streamwise velocity contours induced by the perturbation applied at \(x=x_0=10\); only a truncated version of the larger computation domain is shown.

The inviscid modes are comparable to those determined by Hall and Horseman [23] in the isothermal case for Görtler vortices and those of Andersson et al. [21] for boundary-layer streaks. This is to be expected as the qualitative downstream structure of our thermally induced streaks is comparable, and the inviscid disturbances remain decoupled from thermal effects.

We have examined the Falkner–Skan flow over a flat plate, where the plate is subjected to a localised heating on a spanwise scale of \(O(\text {Re}^{-1/2})\); the simplest way to achieve this is via a localised surface temperature ‘hot spot’ such as (34). The Grashof number based on the peak hot spot temperature is assumed to be \(O(\text {Re}^{3/2})\) and this is sufficient to provide buoyancy forcing of the transverse momentum within the boundary layer. In Sect. 3, we have determined the solution downstream of such a localised hot spot, showing that the perturbation to the streamwise velocity grows algebraically as \(x^\lambda \) (\(\lambda >0\)) when the thermal forcing remains weak (Fig. 5). The linear form of the downstream algebraically growing disturbance is obtained directly in Sect. 2 via an eigenproblem. For stronger forcing, or equivalently further downstream, the perturbation amplitude becomes comparable to the two-dimensional boundary-layer base flow and a three-dimensional nonlinear streak is obtained (Fig. 8). In this strongly nonlinear regime the streak has been shown to be inviscidly unstable to Rayleigh waves (Figs. 7 and 8), which grow exponentially downstream with growth rate \(O(\text {Re}^{1/2})\). Hence, even modest localised temperature perturbations can force a streak response that will ultimately break down. Whilst a localised hot spot with a two-dimensional (x, Z dependent) heating of the type considered here has not been studied experimentally, Dovgal et al. [25] (for example) studied the influence of spanwise uniform heating near the leading edge so similar approaches could be applied here.

The form of the downstream algebraically growing disturbance is obtained using a bi-global eigenvalue computation, for which the non-parallel development of the modes is captured by similarity variables for the transverse and spanwise coordinates. The dominant eigenmode for the Blasius boundary layer in air (Prandtl number \(\text {Pr}\approx 0.71\)) has \(\lambda _1\approx 0.203\) and the corresponding eigenfunction is as given in Fig. 1a in the cross section. This predicted growth rate of the bi-global eigenvalue problem is in agreement with that obtained in the nonlinear non-self-similar computations of Sect. 3 (Fig. 5) in the linear regime.

The self-similar nature of the eigenvalue problem can also be preserved if the plate is heated at all positions such that its temperature is \(x^{3(n-1)/2}\), which introduces a (redefined) scaled Grashof parameter (\(\text {G}\)) based on the bulk plate temperature. Whilst this restricted spatial dependence is of limited practical interest, we note that even in this case, the eigenvalue spectrum is only weakly affected for \(\text {G}=O(1)\) (Table 1). For any plate temperatures that decay slower than this, we would expect the mechanism of Hall and Morris [16] to dominate.

For small Grashof parameter \(\text {G}\ll 1\), the bi-global eigenvalue spectrum can be obtained by a one-dimensional eigenvalue problem arising from the \(k\ll 1\) Fourier transform of the energy equation (17); k being the spanwise wavenumber in the transformed solution. This one-dimensional eigenvalue problem can be used to efficiently determine the most dangerous spatial growth/decay \(\lambda =\lambda _1\) over the full (\(\beta \)–\(\text {Pr}\)) parameter space, where \(\beta \) is the Hartree parameterisation of the external pressure gradient.

For any Prandtl number, there exists a growing eigenmode embedded in a Blasius boundary layer with \(\beta =0\) (Fig. 3). The spatial growth in this case is bounded between \(0<\lambda _1<0.25\). At more general values of \(\beta >0\) (a Falkner–Skan boundary layer with a favourable pressure gradient) the downstream growth (\(\lambda _1\)) is reduced and above a critical value we obtain decay with \(\lambda _1 < 0\). For air, the dominant spatial growth \(\lambda _1\) remains positive provided that the pressure gradient is less than \(\beta \approx 0.183\).

The eigenproblem can also be examined for adverse pressure gradients (\(\beta <0\)). In the case of \(-0.1988<\beta <0\), it is known [26] that the Falkner–Skan equation (8c) has two branches, which are differentiated by the sign of \(\tau =F''({\eta }=0)\). As expected, the application of an adverse pressure gradient increases the spatial growth of the eigenmodes. As a representative example, with pressure gradient \(\beta =-0.1\), \(\text {Pr}=0.71\), the dominant thermal mode has spatial growth \(\lambda _1=0.307\) for the non-reversed (\(\tau >0\)) Falkner–Skan boundary layer.