Micro-slot injection into a boundary layer driven by a favourable pressure gradient

There has, recently, been increased interest in the use of micro-jets as an active mechanism to control flow separation; see, for example, the experiments of Kumar and Alvi [1] or the theoretical work of van Dommelen and Yapalparvi [2]. These are jets with a low level of injected mass flux, and are driven through a small orifice that is comparable in scale to the boundary-layer thickness. This approach to flow injection is different from the classical description of boundary layers affected by wall injection, which only consider injection over length scales that are significantly longer than the local boundary-layer thickness. Based upon the results of macro-scale injection, mass flux into the boundary layer through a surface is usually assumed to promote boundary-layer separation; see, for example, Catherall et al. [3]. Therefore, the ability of a micro-jet mechanism to inhibit separation in a boundary layer is somewhat counter intuitive, and the reason may lie in a fundamentally different response associated with short-scale spanwise forcing. When the boundary layer is forced over a sufficiently short spanwise length scale, it is necessary to re-introduce the effects of momentum diffusion in the spanwise direction. This leads to a slightly more complex dimensionless boundary-layer formulation: for a steady incompressible flow, where \({\hat{V}}\) is the velocity normal to the boundary and \({\hat{W}}\) is the spanwise velocity. Here (Y, Z) are scaled boundary-layer coordinates in the transverse and spanwise directions, respectively, and \(U_e\) is the (known) outer streamwise velocity such that \(({\hat{U}}, {\hat{W}}) \rightarrow (U_e(x),0)\) as \(Y \rightarrow \infty \); we will derive this system more formally in the next section. As for the classical boundary layer, the leading-order streamwise pressure gradient is determined by the free stream to be \(-U_e(x) U_e^\prime (x)\), but a three-dimensional approach requires the inclusion of a higher-order (unknown) pressure correction \({\hat{P}}\). This system is in essence a parabolised version of the Navier–Stokes equations, and as such can be marched downstream in x provided that there is no flow reversal of the streamwise velocity \({\hat{U}}\). This formulation has been considered before, for example, by Patankar and Spalding [4] and Patankar [5], in which (1) is referred to as ‘three-dimensional parabolic flow’. The same system of equations also appears earlier in the work of Kemp [6] and was termed the ‘boundary-region equations’, a phrase that persists in the later literature, see, for example, the recent work of Goldstein et al. [7, 8] or Ricco and Dilib [9].

The classical two-dimensional boundary-layer equations are simply recovered by seeking solutions of (1) with \(W=0\) (no cross flow) and no dependence on the spanwise coordinate Z. This approach leaves only (1a) and (1d) to determine the two-dimensional velocity field (U, V), whilst (1b) is a decoupled equation that allows one to subsequently determine \({\hat{P}}\). It is well known that this two-dimensional problem admits self-similar states for algebraically developing external flow speeds (that is, \(U_e(x)=x^{n}\)) in the form of the Falkner–Skan family of solutions: Using the change of variableswhere \(F(\eta )\) is a scaled streamfunction, the continuity equation (1d) becomes \(U = F'(\eta )\), and the streamwise momentum equation (1a) reduces to the Falkner–Skan equation:where \(\beta = 2n/(n+1)\) is the Hartree parameter [10].

In the two-dimensional system, a self-similar wall injection is defined by the value of F(0) (which is negative for injection), and a key phenomenon for the case of \(n=0\) (zero pressure gradient) is the so-called ‘blow-off’ event described by Kassoy [11]. In the absence of a pressure gradient, there is a critical (negative) value for F(0), at which the boundary-layer solution terminates (that is, it is ‘blown off’ the boundary). Upon approaching this value of the injection rate, the shear stress at the surface of the plate approaches zero, whilst the displacement of the boundary layer tends to infinity. The introduction of a positive pressure gradient (\(n>0\)) into this same problem removes this singularity, and two-dimensional boundary-layer solutions continue to exist for all injection rates but with the increasing displacement as the injection rate increases. The relationship between the zero and non-zero pressure-gradient cases was clarified in a later paper by Kassoy [12] and also discussed in a non-interactive framework by Watson [13]. An obvious issue to address is the relevance of these classical (macro-scale) injection features to problems that involve micro-scale (Z-dependent) injection.

The recent work of Hewitt et al. [14] extended the classical two-dimensional formulation, in the absence of a pressure gradient (i.e. \(n=0\) such that \(U_e=1\)), to a three-dimensional formulation by allowing injection over a spanwise length scale comparable to the boundary-layer thickness. This provides a new family of three-dimensional self-similar solutions to (1) that are analogues of (2) (albeit for \(n=0\)) and driven by injection over short spanwise scales. Counter to the two-dimensional theory, Hewitt et al. showed that the resulting three-dimensional solutions continue to exist for injection velocities greater than the critical two-dimensional ‘blow-off’ injection rate. These new solutions take the form of low-speed streamwise-aligned streaks. These streak solutions were categorised into one of three regimes, where the geometry of the streak is determined by the injection rate and the width of the injection slot.

In this work, we again examine the effect of short-scale injection in the context of (1), with free-stream speeds of \(U_e(x)=x^n\) and a focus on a favourable pressure gradient \(n>0\). Generalising the three-dimensional boundary-layer injection solutions of [14] to include an applied pressure gradient is important not only because non-uniform external flows are commonplace, but also because we know that for macro-scale injection, the flow character is significantly affected by the pressure gradient.

Although this work begins from (1), we subsequently make use of a transformed version of this system for the computational work that follows. Numerical calculations of the three-dimensional boundary-layer solutions are then presented; illustrative examples of both the zero and non-zero pressure-gradient cases will be shown for comparison. We demonstrate that a favourable pressure gradient significantly changes the flow response, largely eliminating the streamwise streaks found for uniform external flows. We also show that increasingly wide injection slots do not connect smoothly to the known results of the Falkner–Skan solutions when the pressure gradient parameter is below a critical value. Finally, we investigate the role of spatially unstable eigenmodes near to the centreline of the injection slot and relate these to the numerical results obtained for increasing injection-slot width.

We consider a dimensional system of Cartesian coordinates \((x^*,y^*,z^*)\) where \(x^*\) is increasing in the streamwise direction, \(y^*\) is the transverse coordinate and \(z^*\) is the spanwise coordinate. The corresponding velocity field is \((u^*,v^*,w^*)\) and the pressure is \(p^*\); in what follows an asterisk will be used for dimensional quantities. A flat plate defined by \(y^*=0\), \(x^* > 0\) is aligned with the free-stream flow, \(u^*=U^*_e(x^*)\), and the leading edge is located at \(x^*=0\) as shown in Fig. 1. We restrict attention to power-law external flows of Falkner–Skan type with \(U_e^*(x^*) = U^*_\infty (x^*/L^*)^{n}\) where \(U^*_\infty \) is a constant velocity scale, n is the constant pressure-gradient parameter and \(L^*\) is an arbitrary streamwise length scale. By defining a global Reynolds number via \(Re=U^*_\infty L^*/\nu ^*\), where \(\nu ^*\) is the constant kinematic viscosity of the fluid, we can introduce the corresponding boundary-layer expansions for \(Re\gg 1\):

where \(\rho ^*\) is the constant density and \(U_e=x^n\) is the dimensionless free-stream speed. Substitution of (3) into the Navier–Stokes equations leads to the leading-order system of (1). It is worth highlighting that whilst p(x) does not appear at leading order, the next correction \({\hat{P}}\) does.

At the surface of the plate, we impose no-slip boundary conditions \({\hat{U}}={\hat{W}}=0\) and an injection velocity \({\hat{V}}={\hat{V}}_\mathrm{{in}}(x,Z)\). It is this injection over the short spanwise scale (Z) that drives the three-dimensional response in the boundary layer. A computationally expensive approach to a discussion of (1) is to exploit the parabolicity in x and march downstream from an initial condition at the leading edge (\(x=0\)), at each location solving for \(({\hat{U}},{\hat{V}}, {\hat{W}}, {\hat{P}})\) in the plane spanned by (Z, Y). In this work, we will take a simpler approach, by looking for three-dimensional solutions that are self-similar in the downstream coordinate. Although this approach is clearly more restrictive, it allows us to clarify the mechanisms that dominate the flow response, in the hope that these same mechanisms still play a similar role in the more general (developing in x) flow.

Key to a self-similar solution is the new coordinate system This coordinate \(\eta \) is the usual Falkner–Skan similarity variable of (2a). To retain spanwise diffusion, at all downstream positions, we require that the spanwise lengthscale is always comparable to the transverse thickness of the layer, hence we employ the same form for \(\zeta \). For the velocity components, we seek a three-dimensional analogue of the standard solutions (2b):where

We apply (4) to the leading-order ‘boundary-region equations’ (1), but first it is convenient to cross differentiate (1b) and (1c) to eliminate the pressure correction \({\hat{P}}\). This cross differentiation naturally introduces a modified vorticity component; we will denote this vorticity by \({\varTheta }\), following the notation of Pal and Rubin [16]. At any x location, the similarity solution is determined over the cross-sectional \((\zeta ,\eta )\) plane by finding the four unknowns \((U,{\varPhi },{\varPsi },{\varTheta })\), which satisfy and where \(\nabla ^2\) is the two-dimensional Laplacian in the \((\zeta ,\eta )\) plane. Despite being somewhat less intuitive than the primitive variable formulation, this form of (1) is well known in relation to corner boundary-layer flows, see for example the formulation of Dhanak and Duck [15] or Pal and Rubin [16].

We will follow the formulation of [15] further by combining (5a) and (5b) to obtain expressions for the Laplacian of both \({\varPhi }\) and \({\varPsi }\) given by This approach simplifies the numerical discretisation.

Our task is to solve (6) for \((U,{\varPsi },{\varPhi },{\varTheta })\) in the \((\zeta ,\eta )\) plane, at which point a three-dimensional solution is obtained at any point in the flow field via (4).

When there is no injection through the surface of the plate, \(K=0\), the Falkner–Skan solution given by (2) is recovered and the flow remains two-dimensional throughout the domain. However, any non-zero injection through the plate surface over a finite spanwise extent \(\zeta _0\) generates a three-dimensional deviation from the two-dimensional base flow in this neighbourhood. We present results for \(K>0\) (injection) and \(0\le n < 1\), corresponding to \(0\le \beta < 1\); when sufficiently far from the injection slot, such that the injection effects are small, \(\beta =0\) leads to a Blasius flow whilst \(\beta \rightarrow 1\) leads to a Hiemenz flow. In this range of pressure gradients, the boundary-layer thickness increases downstream from the leading edge.

The previous work of [14] showed that for \(K>0\) (injection) and no applied pressure gradient (\(\beta =0\)), three distinct flow regimes are observed, as the injection-slot width \(\zeta _0\) is increased. These three regimes are delineated by the injection rates: \(0< K <K_I\approx 0.876\), \(K_I<K<K_{II}\approx 1.95\) and \(K>K_{II}\). The left-hand column of Fig. 2 shows the three flow regimes present when \(\beta =0\). In the weak injection regime (a) \(0<K <K_I\), although a three-dimensional state is obtained, the solution remains qualitatively similar to the underlying Blasius solution with a weak spanwise variation. Near to the centreline \(\hat{\zeta }=0\), there is a slight thickening of the boundary layer, but the injection flow does not have a significant impact on the solution. Upon increasing the injection amplitude such that \(K_I< K < K_{II}\), a low-speed streak develops in the injection region \(\hat{\zeta }< 1\). In this moderate injection regime (c), a displaced shear layer separates the low streamwise speeds inside the streak from the high-speed outer flow. The size of this streak remains confined to lie within the slot region (i.e. \(\zeta < \zeta _0\)). In the strong injection regime (e) \(K > K_{II}\), a much more prominent low-speed streak is observed which is separated from the outer flow by a semi-circular free shear layer. Due to the low streamwise velocity inside the streak, the mass injected into the layer through the plate must be ejected through the shear layer, and this constraint allows one to predict the streak radius as \(\zeta _0\) increases [14].

In the presence of a favourable pressure gradient (\(\beta >0\)), Fig. 2b, d, f, we do not see the three distinct types of flow (as found for \(\beta =0\)) with their prominent low-speed streaks. The introduction of a non-zero pressure gradient, as shown in the right-hand column of Fig. 2 (with \(\beta =0.1\)), significantly alters the streamwise velocity U in the neighbourhood of the slot (\(|\zeta |<\zeta _0\)). The two-dimensional Falkner–Skan solution that exists far away from the injection slot \((\hat{\zeta }\gg 1)\) is modified in the injection region to a much lesser degree for \(\beta >0\). There is no obvious semi-circular low-speed region as shown by comparison of Fig. 2e for \(\beta =0\) and Fig. 2f for \(\beta =0.1\). In general, injection leads to a thickening of the near-plate layer, which is to be expected.

Instead, we obtain a much less dramatic flow response, but one that nonetheless has two distinct properties: we either find that at fixed \(K>0\) (injection) and \(\beta >0\) (i) the solution near the centreline becomes increasingly displaced from the plate as \(\zeta _0\) increases or (ii) the low-speed region in the slot region \(\zeta <\zeta _0\) remains largely unaffected as the slot width is increased. This behaviour is seen in Fig. 3a, b where contours are shown for \(K=4\), \(\beta =0.2\) and \(\zeta _0=20\) in (a) and for \(\zeta _0=40\) in (b). However, this growing \(\eta \) scale is not observed in all cases, as demonstrated by Fig. 3c, d in the case \(K=4\), \(\beta =0.8\), \(\zeta _0=20\) in (c) and \(\zeta _0=40\) in (d). We will return to explain this feature in the subsequent analysis below, to identify how the choices of \(\beta \) and K determine the observed behaviour.

Comparing these results to the zero pressure-case (\(\beta =0\)) we find that a favourable pressure gradient acts to inhibit (although not eliminate entirely) the rapid displacement of a low-speed layer from the plate surface. The greatly reduced extent of the low-speed streak indicates an increase in the streamwise mass flux and therefore a reduced radial flow into the far field, compared with the zero pressure-gradient case. This reduction in the radial mass flux is reflected in a reduction in the magnitude of the mass flux parameter A contained in (12). Again, we return to validate this feature via asymptotic methods below.

An asymptotic description of the flow in the limit of a large injection-slot width, \(\zeta _0 \rightarrow \infty \), is presented below. When \(0< \beta < 1\) a large aspect ratio viscous layer is present, spanned by \(\eta =O(1)\) and \(\zeta =O(\zeta _0)\) for all values of \(K>0\). To describe the flow, we use the rescaled coordinate \(\hat{\zeta }= \zeta / \zeta _0\) and Using (13) and taking the limit \(\zeta _0 \rightarrow \infty \), we find that at leading order, (6) reduces to This system is parabolic in \(\hat{\zeta }\) and the ‘windward’ direction is determined by the sign of \({\varPsi }_0\). If \({\varPsi }_0 > 0\) for all \(\eta > 0\) then the far-field solution can be extended from \(\hat{\zeta }\gg 1\) (where the flow is know to be the classical Falkner–Skan solution) towards \(\hat{\zeta }=0\) by parabolic marching, subject to the conditions where \(V_\mathrm{{in}}\) is defined by (8).

In the far-field \(\hat{\zeta }\gg 1\), the ‘initial’ state for (14) is the Falkner–Skan solution, for which \({\varPsi }_0 = (1-\beta ) U_0\), which means that there is no cross flow so \(W=0\) in (4d). If we start (14) from a state with no cross flow, then it retains this feature for all subsequent \(\hat{\zeta }\), and hence the system (14) can be reduced to a simpler form with \(U_0=0\), \({\varPhi }_0 = -V_\mathrm{{in}}\) on \(\eta =0\) and \(U_0 \rightarrow 1\) as \(\eta \rightarrow \infty \). We may also note that setting \(\hat{\zeta }=0\), under the assumption that \(\hat{\zeta }\) derivatives can then be neglected in this limit, reproduces the Falkner–Skan equation (2d), but subject to an injection at the surface of \(V_\mathrm{{in}}(\hat{\zeta }=0)=K\). This suggests that for sufficiently large slot widths, we may eventually recover the classical Falkner–Skan solution at the centreline of the injection region. However, this is not always true, and as we shall see in the next section, small amounts of residual cross flow can dominate for small \(\hat{\zeta }\), leading to rather different behaviours in some cases. Nevertheless, we begin by assuming that such cross flow is small and that (15) is sufficient to describe the flow.

When \(\beta =0\) solutions to (2d) only exist for \(K \lesssim 0.876\). Marching of (15) for \(K \gtrsim 0.876\) leads to a singular response at a finite value of \(\hat{\zeta }>0\) because the wall shear stress \(U_{0\eta }(\eta =0)\) becomes equal to zero; this is associated with a change in the ‘windward’ direction of the parabolic system. When \(\beta > 0\), marching of (15) to \(\hat{\zeta }=0\) can be achieved for all values K. It is the presence of this singularity (when \(\beta =0\)) that leads to the other types of low-speed streaks described by Hewitt et al. [14].

The behaviour of the downstream shear is illustrated in Fig. 4. When \(\beta \ne 0\) we can see that the wall shear stress never reaches zero and instead approaches a finite value as \(\hat{\zeta }\rightarrow 0\). For large injection rates K, we find that the wall shear \(U_{\eta }(\eta =0) \sim \beta / K\) in the injection region thus agreeing with the two-dimensional theory of Kubota and Fernandez [21].

For large \(\eta \), the vertical velocity component in the parabolic solution is of the formHere \(\delta _{F}\) is the constant displacement thickness associated with the Falkner–Skan solution for flow on a flat plate in the absence of injection. We also have a displacement \(\delta _\mathrm{{in}}(\hat{\zeta })\) which is varying in the spanwise direction due to the wall injection over a finite spanwise region. If there is no injection though the plate surface then \(\delta _\mathrm{{in}}\equiv 0\) for all \(\hat{\zeta }\) and there is no \(\hat{\zeta }\)-dependent perturbation to the Falkner–Skan solution, however in general \(\delta _\mathrm{{in}}(\hat{\zeta })\) is non-zero and must be taken into consideration.

We now consider the inviscid region defined by \(\eta = \zeta _0 \hat{\eta }\) and \(\zeta = \zeta _0 \hat{\zeta }\). In this outer region, we may rewrite the vertical velocity component aswhere \(\phi =O(1)\) in order to match with the spanwise varying displacement induced by the parabolic solution (15) and \(\zeta _0 \hat{\eta }+ \delta _F\) is the (outer) \(\eta \rightarrow \infty \) limit of the (inner) Falkner–Skan solution. Rewriting (5a) and (5b) in terms of \(\hat{\zeta }\) and \(\hat{\eta }\), and since both U and \({\varTheta }\) remain \(o(\zeta _0^{-2})\), we find that \(\phi \) is determined by the harmonic problem, subject to Here \(\delta _\mathrm{{in}}(\hat{\zeta })\) is determined by the solution of the parabolic system (15). It is also possible to obtain a similar Laplace problem for the corresponding \({\varPsi }\) velocity perturbation \(\psi (\hat{\zeta },\hat{\eta })\). The solution to the problem (18) is obtained using a Green’s function approachWe know from (12) that in the far fieldwhere \(A = \zeta _0 \hat{A}\). Given this far-field behaviour, the total radial mass flux due to the spanwise variation at the edge of the boundary layer is \({\hat{A}}\zeta _0 \pi = A \pi \). On the other hand, the flux from the inner parabolic region induced by the injection is \(M \zeta _0\) whereCombining these two expressions for the mass flux givesso the mass flux coefficient A is proportional to the blowing width \(\zeta _0\) and the integral of the spanwise varying displacement. The description provided by (22) applies to the \(\beta =0\) case of [14], but the difference here lies in the mass flux integral being a function of the pressure-gradient parameter, \(M=M(\beta )\).

This asymptotic description suggests that, for large injection slot widths \(\zeta _0\gg 1\) and \(\beta > 0\), the mass flux coefficient A remains linearly dependent upon \(\zeta _0\) for any fixed \(K>0\). This is markedly different to when \(\beta = 0\), in which case increased injection \((K \gtrsim 1.95)\) leads to a mass flux coefficient A that is proportional to \(\zeta _0^2\) and the formation of a low-speed semi-circular ‘bubble’.

Computation of \(\delta _\mathrm{{in}}(\hat{\zeta })\) by parabolic marching of (15), allows one to determine \(M(\beta )\) via the integral (21). In Fig. 5, we show the asymptotic prediction (22) for a range of values of \(\beta =0.05,0.1,0.4,0.8\), which effectively shows the reduction in radial mass flux in the \((\zeta ,\eta )\) plane as the pressure gradient is increased. The same figure also shows numerically determined values of A, as obtained from (6), in the particular case \(\beta =0.1\) and increasing injection-slot widths \(\zeta _0=2,4\) and 8. The linear relationship between the mass flux coefficient A and the injection width \(\zeta _0\) is confirmed with good agreement between the numerically determined values and the asymptotic prediction for the representative value of \(\beta =0.1\).

The leading-order parabolic solution of (15) captures the full numerical solution for \(\hat{\zeta }=O(1)\), as evidenced by the results of Figs. 4 and 5, but there are some additional features that it does not describe. One feature is the influence of spanwise diffusion around the edge of the injection slot, although this region is small compared to the \(\hat{\zeta }=O(1)\) scale. A more important issue is the behaviour shown in Fig. 3. For some choices of \(\beta \) and K, we find an increasing boundary-layer thickness local to the centreline of the slot as \(\zeta _0\) increases. We now examine the role of cross-flow eigenmodes to the parabolic system (15), which allows us to determine a critical favourable pressure gradient beyond which this thickening local to the centreline is not found.

In Fig. 6, we show the streamwise velocity profiles as measured on the centreline of the injection slot (\(\hat{\zeta }=0\)) for \(K=2\), \(\beta =0.1,0.8\) and \(\zeta _0=20,40\). When \(\beta =0.8\) the parabolic solution (15), in the limit of \(\hat{\zeta }\rightarrow 0\), is in agreement with the numerical solution of the full equations (6) for both \(\zeta _0=20\) and \(\zeta _0=40\). On the other hand, when \(\beta =0.1\) the parabolic solution and the full solution only agree near to \(\eta =0\) and for large \(\eta \), whilst at intermediate values of \(\eta \) there is a large discrepancy. Furthermore, increasing the slot width (\(\zeta _0\)) only exacerbates the difference. This reinforces the picture of the flow response presented by Fig. 3, which similarly shows a benign dependence on \(\zeta _0\) when \(\beta =0.8\) compared to the eruptive behaviour found for \(\beta =0.2\). This deviation from the behaviour predicted by the parabolic solution is confined to near the centreline \(\hat{\zeta }\ll 1\) and as such it is necessary to alter our approach in this region. It is clear that, as the slot is widened, some pressure gradients result in a centreline flow that is consistent with the corresponding injection-affected Falkner–Skan solution, but some instead result in what appears to be an increasing eruptive behaviour in this region.

In order to understand the behaviour near to \(\hat{\zeta }=0\), we seek to describe the spatial evolution of a small perturbation to the parabolic solution of the previous section. To perform this, we expand in the form Setting \(\epsilon =0\) we recover the leading-order system (14), which if started from a state with no cross flow, has a solution with \({\varPsi }_0=(1-\beta )U_0\), which leads to the simpler form (15). For \(\epsilon \ll 1\), the linearised system for \((u,\varphi ,\psi ,\vartheta )\) is given by For \(\hat{\zeta }\ll 1\), solutions to these \(O(\epsilon )\) perturbation equations exist in the formThe \(O(\epsilon ^0)\) system merely reduces to the Falkner–Skan equation (2d) with an injection boundary condition such that \(U_0=F'(\eta )\), \({\varPhi }_0=F(\eta )\), \({\varPsi }_0 = (1-\beta )F'(\eta )\), \({\varTheta }_0=(1-\beta )F''(\eta )\) and \(F(0)=-K\), along with the other usual boundary conditions. The \(O(\epsilon )\) system, with the local form (25) leads to an eigenvalue problem where the prime notation is used to denote derivatives with respect to \(\eta \). The homogeneous boundary conditions to be imposed are \(u_0(0)=\varphi _0(0)=\psi _0(0)=0\) and \(u_0,\psi _0,\vartheta _0 \rightarrow 0\) as \(\eta \rightarrow \infty \). Here \(\lambda = \lambda _r + i \lambda _i\) is a complex eigenvalue which we must determine, to describe the spatial behaviour of the perturbation on approaching the slot centreline. If the real part of the eigenvalue is negative \((\lambda _r < 0)\) as \(\hat{\zeta }\rightarrow 0\) then the perturbation grows and the corresponding eigenmode is linearly unstable. In such cases, we should not expect, in general, to recover a Falkner–Skan solution near \(\hat{\zeta }=0\).

Using a standard central differencing scheme, we may discretise (26) producing a generalised eigenvalue problem of the form \(A \mathbf {v}_n = \lambda _n B \mathbf {v}_n\) where A and B are \(6N_{\eta } \times 6N_{\eta }\) matrices and \(n \in \{1,\ldots , 6N_{\eta }\}\). Performing a generalised Schur decomposition to factorise A and B, we are able to find values of \(\lambda _n\) which are numerical approximations to the eigenvalue spectrum of (26).

Figure 7a shows the real part of the most relevant eigenvalue for various values of the ‘injection’ parameter K with \(\beta =0.1,0.2,0.4,0.5,0.6\). For a more complete picture, the figure covers \(K\in [-2,3]\) with \(K<0\) indicating suction and \(K>0\) injection. The most relevant eigenvalue in this case is the one with the smallest real part, and in Fig. 7a, the eigenvalues shown are all purely real with \(\lambda _i=0\).

When \(\beta =0.1\) (for example) the smallest eigenvalue is positive when K is sufficiently negative (suction) but as K increases the real part of the eigenvalue becomes negative, \(\lambda _r < 0\), and approaches a constant negative value for large K (strong injection). With \(\lambda _r < 0\) this eigenmode will come to dominate the solution as \(\vert \hat{\zeta }\vert \rightarrow 0\), and when sufficiently close to the centreline of the injection slot one must appeal to the re-inclusion of spanwise diffusion and/or nonlinear effects to regularise this linear spatial growth. The eigenfunction of this spatial mode is such that there is a non-zero cross-flow velocity associated with the perturbation, that is \(W\ne 0\) as defined by (4d).

Figure 7a demonstrates the impact of increasing the pressure gradient on the spatial stability of the solution near to \(\hat{\zeta }=0\). As the value of \(\beta \) increases, \(\lambda _r\) typically increases until at a critical value \(\beta _\mathrm{{c}}(K)\) the eigenvalue with the smallest real part has \(\lambda _r \ge 0\). This means that only for \(\beta \ge \beta _\mathrm{{c}}(K)\) can we expect to recover the Falkner–Skan solution for \(\vert \hat{\zeta }\vert \ll 1\) if there are perturbations with non-zero cross flow.

Figure 7b shows the functional relationship between injection and the pressure gradient parameter, such that there exists a spatially neutral mode with \(\lambda _r=0\); as such this represents a neutral curve above which we expect a benign response near to \(\hat{\zeta }=0\) and below which we expect the solution to be affected by these centreline eigenmodes. As seen in Fig. 7a, the eigenvalue approaches a constant value for sufficiently large \(K>0\) (strong injection). Therefore, we expect \(\beta _\mathrm{{c}}\) to also asymptote to a constant as K increases. As shown in Fig. 7b, the numerical evidence suggests that \(\beta _\mathrm{{c}} \approx 1/2\) for large \(K>0\), although we have not sought to demonstrate this explicitly via an asymptotic analysis.

Three-dimensional similarity solutions have been presented for laminar flow of velocity \(O(U^*_\infty x^n)\) over a flat plate, driven by a favourable pressure gradient that is parametrised by n in the usual Falkner–Skan approach. The three dimensionality is driven by injection over the short spanwise scale of width \(O\left( \zeta _0 Re^{-1/2} x^{(1-n)/2} L^* \right) \). The injection velocity is weak, \(O\left( K Re^{-1/2} x^{(n-1)/2} \right) \), where x is a dimensionless streamwise coordinate and Re a Reynolds number based on the streamwise length scale \(L^*\). The above choices for the slot and injection velocity scaling mean that the slot width is always a constant ratio of the boundary-layer thickness at all downstream locations. This approach allows for solutions that are self-similar in the downstream x coordinate, being three-dimensional extensions of the Falkner–Skan solution. In general, this need not be the case, and the approach can be generalised to parabolic marching in the downstream direction to account for other choices. Away from the injection slot, at large spanwise distances, the flow returns back to the classical two-dimensional Falkner–Skan profile appropriate for an impermeable boundary.

In the absence of a free-stream pressure gradient (that is \(\beta =0\), where \(\beta \) is the Hartree parameter), the earlier work of Hewitt et al. [14] has shown that an increasing injection-slot width (\(\zeta _0\)) leads to one of three flow regimes. Selection between these three regimes depends upon the size of the injection velocity which is parametrised by K. The two critical values of \(K>0\) that separate the three regimes are \(K=K_I \approx 0.876\) and \(K=K_{II} \approx 1.95\). The three flow regimes are exemplified by the streamwise velocity contours shown in the left hand column of Fig. 2. However, in the presence of a favourable pressure gradient (\(\beta > 0\)) these three distinct flow responses cease to exist and the low-speed streak structures present in the second and third regimes very rapidly disappear, as shown in the right-hand column of Fig. 2 for \(\beta =0.1\). The removal of streamwise-aligned streaks greatly reduces the radial flow into the far field in the (\(\zeta ,\eta \)) plane, corresponding to a reduction in the magnitude of the mass flux parameter A defined by (12), which we demonstrate must be such that \(A=O(\zeta _0)\) for \(\beta >0\).

Far from the injection slot (\(\hat{\zeta }\gg 1\)) the flow remains unaffected by the (\(\hat{\zeta }=O(1)\)) injection-slot region, resulting in a two-dimensional solution of Falkner–Skan type. This solution has no cross flow at leading order (\(O(\zeta _0)\)) and can be continued from large \(\hat{\zeta }\) towards \(\hat{\zeta }=0\) by parabolic marching of (15). For sufficiently favourable pressure gradients, \(\beta _\mathrm{{c}}(K)<\beta < 1\), this results in good agreement with the full numerical solution of (6) outside of a viscous-dominated layer that straddles the edge of the injection region. For weakly favourable pressure gradients \(0<\beta <\beta _\mathrm{{c}}(K)\), we still find good agreement with the numerical solution, but this agreement fails dramatically near the centreline of the injection slot.

This study therefore highlights some fundamental qualitative differences associated with the presence of both short-spanwise scales and a favourable streamwise pressure gradient. For a strongly favourable pressure gradient, above the critical value \(\beta _\mathrm{{c}}(K)< \beta < 1\), any residual cross flow does not grow in the injection region and as the slot is widened, one simply recovers the classical Falkner–Skan solution local to the centreline. However, for \(0<\beta <\beta _\mathrm{{c}}(K)\) any residual cross flow becomes increasingly important on approaching the centreline. Mathematically this behaviour arises through the appearance of a spatially unstable eigenmode as \(\hat{\zeta }\rightarrow 0\). Crucially the eigenfunction of this ‘unstable’ mode has non-zero cross flow, and so it is only triggered by the higher-order corrections for large slot widths. A practical consequence of this behaviour is that the limit of wide injection slots in this ‘micro-slot’ formulation only recovers the Falkner–Skan solution for \(\beta _\mathrm{{c}}(K)<\beta <1\). If \(0<\beta <\beta _\mathrm{{c}}(K)\) an increasingly wide injection slot results in a cross-flow entrainment of fluid from outside the injection slot and its transport towards the centreline of the slot. For \(0<\beta <\beta _\mathrm{{c}}(K)\) this cross flow leads to collisional behaviour along the centreline of the slot, which is in turn associated with a thin \(\hat{\zeta }\ll 1\) eruption region as the slot widens (\(\zeta _0\gg 1\)).