1 Introduction and formulation

In a Blasius boundary layer (and similar shear flows) it is well known that streamwise vorticity is highly efficient at generating streamwise streaks. At high Reynolds numbers, the asymptotically weak ‘roll’ motion associated with streamwise vorticity moves low-speed fluid from the wall into regions of higher streamwise velocity, with a return flow of higher-speed fluid towards the wall. This has been recognised for some time as a source of $O(1)$ low/high-speed streaks aligned with the free-stream direction. In the non-parallel context of the Blasius boundary layer, the in-plane roll motion and the downstream streak remain fully coupled, and herein we use the term ‘streak’ to indicate this three-dimensional disturbance on an otherwise two-dimensional Blasius boundary layer.

The role of streamwise streaks/vortices in transient growth has been extensively studied, see for example the discussion of Schmid & Henningson (Reference Schmid and Henningson2012). For boundary layers it is known (Andersson, Berggren & Henningson Reference Andersson, Berggren and Henningson1999) that the greatest amplification is associated with streaks that have spanwise scales comparable to the transverse boundary-layer thickness. The streak states that arise (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001) provide a primary structure on which secondary instabilities arise as the amplitude of the streak is increased. Of course the Blasius boundary layer is also unstable via the classical route of Tollmien–Schlichting (TS) waves, for sufficiently large Reynolds numbers. There is also evidence that artificially forcing streak states can stabilise a boundary layer to TS waves (Cossu & Brandt Reference Cossu and Brandt2002), so long as the streak amplitude remains below that required for a secondary instability.

Much of the analysis on ‘streaky’ boundary layers assumes periodicity in the spanwise direction. There are obviously some computational advantages to this assumption as well as empirical evidence of the importance of such states in flows disturbed by free-stream turbulence (for example). There are also a number of experimental investigations in which the disturbance is driven by spanwise-periodic distributions of wall roughness, injection or suction; see for example Bakchinov et al. (Reference Bakchinov, Grek, Klingmann and Kozlov1995) and White (Reference White2002). A notable exception is the experimental work of Asai, Minagawa & Nishioka (Reference Asai, Minagawa and Nishioka2002), in which a single streak perturbation is induced as a primary state, with the secondary instability induced via small harmonic oscillations imposed by injection through holes in the boundary surface.

Our focus in this work is spatially localised (rather than periodic) streaks, driven by wall forcing over a finite spanwise and streamwise extent; we begin with a theoretical linearised description before presenting analogous nonlinear streak states. Before developing our approach it is instructive to review some related theoretical results.

Linearised perturbations to a Blasius boundary layer have been considered in many contexts. Of particular relevance is the work of Libby & Fox (Reference Libby and Fox1963), which identified a set of two-dimensional (steady) eigenmodes that exist on a Blasius base state. The Libby & Fox modes are governed by the two-dimensional leading-order boundary-layer equation

where $\unicode[STIX]{x1D6F7}(x,Y)$ is a streamfunction, $x$ is a downstream coordinate and $Y$ is a rescaled boundary-layer coordinate. A solution exists in the form

which then provides an evolution equation for perturbations $f$ of amplitude $\unicode[STIX]{x1D716}$ to the Blasius base state defined by $F$ , which satisfies

The perturbation $f(x,\unicode[STIX]{x1D702})$ represents a downstream evolving (linear, if $\unicode[STIX]{x1D716}\ll 1$ ) perturbation. As shown by Libby & Fox (Reference Libby and Fox1963), a linearised solution for $f$ exists as a superposition of eigenmodes. These modal solutions take the form of $f(x,\unicode[STIX]{x1D702})=x^{-\unicode[STIX]{x1D70E}}\bar{f}(\unicode[STIX]{x1D702})$ , giving rise to a one-dimensional eigenvalue problem for the spatial (algebraic) decay ‘rate’ $\unicode[STIX]{x1D70E}$ . The least-damped mode has $\unicode[STIX]{x1D70E}=1$ , for which an analytical solution was previously noted in the work of Stewartson (Reference Stewartson1957).

A closely related approach was later taken in the work of Luchini (Reference Luchini1996) albeit with a different emphasis. The focus of this later work is on three-dimensional modes, initially formulated for spanwise length scales that are comparable to the boundary-layer thickness, although a large spanwise wavelength limit is then taken. The goal was to gain insight into the transient growth mechanism associated with streak perturbations. The follow-up work of Luchini (Reference Luchini2000) went on to consider disturbances with a spanwise wavelength comparable to the boundary-layer thickness (including so-called optimal disturbances).

To place this three-dimensional formulation into a specific context, we consider a semi-infinite flat plate aligned with an oncoming incompressible, uniform flow of speed $U_{\infty }^{\ast }$ . An arbitrary choice of reference length scale, $L^{\ast }$ , allows for a non-dimensional Cartesian coordinate system $(L^{\ast }x,L^{\ast }y,L^{\ast }z)$ aligned with the leading edge of the plate at $x=0$ , such that $y=0,x>0$ defines the plate surface. To capture short spanwise scales typical of streaks we rescale in the $(y,z)$ plane according to

Here $Re=U_{\infty }^{\ast }L^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ (for kinematic viscosity $\unicode[STIX]{x1D708}^{\ast }$ ) is a global Reynolds number based on the chosen length scale. The corresponding high Reynolds number flow field (assumed to be steady) is

with pressure

For large Reynolds number the solution is therefore governed by

This system is often referred to as the boundary-region equations, terminology that is attributed to Kemp (Reference Kemp1951). It is to be solved subject to no-slip constraints on the plate ( $Y=0$ ) and matching to a uniform free stream in the far field. It has been previously employed in discussions of (spanwise-periodic) streaks (Higuera & Vega Reference Higuera and Vega2009), their transient growth (Luchini Reference Luchini1996, Reference Luchini2000; Andersson et al. Reference Andersson, Berggren and Henningson1999) and their self-sustained interaction with travelling waves (Hall & Sherwin Reference Hall and Sherwin2010). Indeed, comparable governing equations have been employed in many other less directly related applications (including Rubin Reference Rubin1966; Hall Reference Hall1988; Dhanak & Duck Reference Dhanak and Duck1997; Goldstein et al. Reference Goldstein, Sescu, Duck and Choudhari2016; Hewitt, Duck & Williams Reference Hewitt, Duck and Williams2017; Williams & Hewitt Reference Williams and Hewitt2017).

To tackle (1.6) Luchini (Reference Luchini1996) firstly assumed periodicity in $Z$ for a linear perturbation of the form $(u(x,Y),v(x,Y),w(x,Y))\exp (ikZ)$ added to a Blasius base flow. However, as noted in that work, taking this approach offers ‘no self-similarity, mainly because the two terms in the sum $u_{YY}-k^{2}u$ scale differently’. To avoid this issue, it was further assumed that $Re^{-1/2}\ll k\ll 1$ (implying small spanwise wavenumber), but three-dimensional effects were retained by allowing $kw\sim O(1)$ (the subsequent computational work of Luchini (Reference Luchini2000) corresponds to $k=O(1)$ ). As for the Libby & Fox modes, solutions are again reduced to a one-dimensional eigenvalue problem by seeking separable states, for example $u(x,Y)=x^{-\unicode[STIX]{x1D707}}\bar{u}(\unicode[STIX]{x1D702})$ where $\unicode[STIX]{x1D702}$ is the transverse coordinate defined in (1.2). In this case it is found that $\unicode[STIX]{x1D707}\approx 0.787-1\approx -0.213$ , leading to algebraic growth of this (spanwise-periodic) perturbation. As we shall show, the value of $0.787\ldots$ turns out to be of relevance even to the localised three-dimensional streak eigenmodes discussed herein.

In this work we will not assume periodicity in the $Z$ coordinate, instead we seek solutions that solve (1.6) exactly, by retaining both the $Y$ and $Z$ dependence. In this sense these states are the natural three-dimensional extension of the Libby & Fox eigenmodes. Central to considering spatially localised streaks, we will require that the solution returns to the two-dimensional Blasius state as $|Z|\rightarrow \infty$ .

We first encompass the downstream growth of the transverse and spanwise length scales via a change of variables

The transverse ( $\unicode[STIX]{x1D702}$ ) scale is well known from the Blasius solution (1.1)–(1.3), and for spanwise and transverse diffusion to remain comparable at all downstream positions we rescale $Z$ in the same manner. Similarly, for the velocity components

For a two-dimensional flow (that is, ${\hat{W}}\equiv 0$ ) this formulation exactly mirrors the usual derivation of the Blasius similarity solution in terms a streamfunction $\unicode[STIX]{x1D6F7}$ , resulting in (1.3).

It is helpful to decompose the solution into a Blasius (two-dimensional) base flow governed by (1.3), plus a three-dimensional streak/roll contribution (denoted by tilde):

The quantity $\unicode[STIX]{x1D6E9}$ is related to a vorticity component, and is introduced by a cross-differentiation to remove the pressure term from (1.6). It is worth emphasising that, as can be seen from (1.7), $\unicode[STIX]{x1D701}U-\unicode[STIX]{x1D6F9}$ is a measure of cross-flow and a three dimensional solution with ${\hat{W}}\neq 0$ is achieved for $\tilde{\unicode[STIX]{x1D6F9}}\neq \tilde{U}$ .

The fully nonlinear, three-dimensional parabolic system that represents the boundary-region equations (1.6) is now converted to:

This system is parabolic in $x$ , subject to the following no-slip and symmetry boundary conditions:

On approaching the free stream ( $\unicode[STIX]{x1D702}\gg 1$ with $\unicode[STIX]{x1D701}$ fixed), following Hewitt & Duck (Reference Hewitt and Duck2014),

where $A$ is a constant determined as part of the solution, representing a measure of the far-field mass flux away/towards the streak into the free stream. For large $\unicode[STIX]{x1D701}$ near the boundary surface ( $\unicode[STIX]{x1D702}=O(1)$ ), there is a spanwise-decaying inner solution that must match (10d,e ) to the no-slip conditions at the surface. The details of this region are given in Hewitt & Duck (Reference Hewitt and Duck2014), but will be revisited in a related context in § 2.1. The arguments against algebraic decay towards the free stream made for traditional boundary-layer problems (see for example the discussions of, Brown & Stewartson Reference Brown and Stewartson1965; Hewitt, Duck & Stow Reference Hewitt, Duck and Stow2002) do not apply to the boundary-region equations as the vorticity decay remains exponential.

2 Spatially localised streak eigenmodes

The evolution of small-amplitude localised perturbations is governed by (1.9), under a linearisation assumption that neglects products of the disturbance $(\tilde{\unicode[STIX]{x1D6F7}},\tilde{\unicode[STIX]{x1D6F9}},\tilde{U} ,\tilde{\unicode[STIX]{x1D6E9}})$ . Given the form of (1.9) there are clearly downstream (decaying) disturbances proportional to $x^{\unicode[STIX]{x1D706}}$ . We may therefore seek linearised solutions of the form

which for $\unicode[STIX]{x1D716}\ll 1$ reduces (1.9) to a linear bi-global (generalised) eigenvalue problem for $\unicode[STIX]{x1D706}$ . We discretise the eigenproblem using a standard second-order central-difference discretisation of the $\unicode[STIX]{x1D701}{-}\unicode[STIX]{x1D702}$ plane, using a non-uniform grid, with a subsequent matrix eigenvalue analysis via the SLEPc library of Hernandez, Roman & Vidal (Reference Hernandez, Roman and Vidal2005). This is a substantial computation, with a sparse matrix eigenproblem of size $4N_{\unicode[STIX]{x1D702}}N_{\unicode[STIX]{x1D701}}\times 4N_{\unicode[STIX]{x1D702}}N_{\unicode[STIX]{x1D701}}$ , where $N_{\unicode[STIX]{x1D702},\unicode[STIX]{x1D701}}$ represents the number of nodal points in each direction.

If we were to ignore the $\unicode[STIX]{x1D701}$ dependence in (2.1) then we should expect to recover the (two-dimensional) eigenmodes of Libby & Fox (Reference Libby and Fox1963) outlined in (1.2). However, these do not satisfy our far-field requirement that modes are localised, i.e. they do not decay with $\unicode[STIX]{x1D701}$ . Nevertheless any eigenmode solutions to the linearised form of (1.9) may be viewed as new three-dimensional (localised) analogues of these classical states.

Table 1.

The first four eigenvalues (all real) obtained from the bi-global computation for domain truncations of the $(\unicode[STIX]{x1D701},\unicode[STIX]{x1D702})$ plane at $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D702}=40,60,80$ .

In table 1 we provide numerical values (to three decimal places) for the first four localised eigenmodes. These values are obtained by fixing the domain size, then increasing the number of mesh points until the first four eigenvalues remain unchanged (to three decimal places) to any further refinement. Only after the eigenvalues are spatially resolved do we then recompute on a larger domain size to assess the influence of domain truncation. The resulting values for $\unicode[STIX]{x1D706}$ are robust to changes in both the number of nodal points in the computational mesh and the truncation of the computational domain. There are no other nearby eigenvalues in the region ( $\unicode[STIX]{x1D706}_{r}>-1.7$ ) spanned by these first four modes and the eigenvalues remain isolated/discrete. We have made no attempt to investigate the behaviour of the faster-decaying modes for $\unicode[STIX]{x1D706}_{r}<-1.7$ .

Figure 1.

The first four localised eigenmodes for a Blasius flow (all real), normalised such that the largest perturbation to $u$ is $-1$ . Contours show the streamwise (streak) velocity component $u(\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})$ , whilst the vector field shows the (in-plane) roll velocity components. These modes correspond to (a) $\unicode[STIX]{x1D706}_{1}$ , (b) $\unicode[STIX]{x1D706}_{2}$ , (c) $\unicode[STIX]{x1D706}_{3}$ and (d) $\unicode[STIX]{x1D706}_{4}$ , as given in table 1. A scale for length of the in-plane velocity vectors is noted at the top of each panel via a vector of magnitude 2 (a–c) and 5 (d).

The values of table 1 correspond to (real) eigenfunctions, which are shown in figure 1 over the $(\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})$ plane. Each subfigure shows contours of the $u$ component of the eigenmode (normalised to make the minimum value $-1$ ) and overlaid is the in-plane ‘roll’ velocity field

Given these results, the perturbation velocity field at any required $x$ location can be obtained merely by rescaling of the $(\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})$ coordinates and the in-plane velocities by $(2x)^{1/2}$ , as specified by (1.7).

It is immediately apparent from figure 1 that some modes decay more rapidly away from the streak, for example figure 1(b). This is particularly noticeable in the near-plate region (e.g. $\unicode[STIX]{x1D702}<2$ ) where in some cases there is a marked cross-flow – that is, a non-zero ${\hat{W}}$ velocity component in the notation of (1.7). In all cases this cross-flow decays to zero for sufficiently large values of $\unicode[STIX]{x1D701}$ ; note that figure 1 shows only a small section of the total computational domain.

3 Nonlinear steady streak/roll states

To compute nonlinear states we undertake a parabolic marching of the full system (1.9) subject to a finite-amplitude disturbance. We choose to force the disturbance via a localised injection through the plate surface:

This ad hoc perturbation will force a streak response with an amplitude parameterised by $\unicode[STIX]{x1D705}$ , and results in the following modified boundary conditions for (1.9)

This model is introduced as conceptually comparable to the experimental configuration of Asai et al. (Reference Asai, Minagawa and Nishioka2002), where a spatially localised streak is generated by a 5.5 mm (or 7.5 mm) (width) by 2.4 mm (height) mesh screen placed perpendicularly to the plate 500 mm from the leading edge. The (undisturbed) displacement thickness at this point is $\unicode[STIX]{x1D6FF}_{0}^{\ast }\approx 2.4~\text{mm}$ . If the downstream response of a perturbation to the Blasius flow is captured by (1.9) then we should also expect to be able to reproduce the general features of these experiments.

We choose a simple, localised disturbance for (3.1) of the form

where $\unicode[STIX]{x1D701}=Z/(2x)^{1/2}$ is a spanwise coordinate measured relative to the local (transverse) length scale of the Blasius solution. This disturbance is localised around $x=x_{0}$ , with decreasing streamwise extent as $\unicode[STIX]{x1D6FE}$ increases. The $(2x)^{1/2}$ factor ensures that the spanwise extent of the disturbance and the amplitude are measured relative to the local Blasius base-flow properties. For the spanwise shape of the disturbance we choose

where $c$ sets the spanwise length scale. To mimic the Asai et al. (Reference Asai, Minagawa and Nishioka2002) experimental configuration the polynomial term is included in (3.3b ) to ensure that there is no net mass flux into the boundary layer, that is $\int _{0}^{\infty }G(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701}=0$ .

Parabolic marching of (1.9) begins at $x=0$ from trivial initial conditions on account of (3.3) being negligible at the leading edge for the considered values of $x_{0}$ . A second-order (Crank–Nicolson) scheme is used to march the system with Newton iteration to handle the nonlinearity. At each $x$ -station and each Newton iteration we are left to solve a $4NM\times 4NM$ sparse system, where $N,M$ are the number of nodes in $\unicode[STIX]{x1D701},\unicode[STIX]{x1D702}$ associated with a (second-order) central-difference discretisation. Each sparse solve is performed using a direct parallel solver (Amestoy, Duff & L’Excellent Reference Amestoy, Duff and L’Excellent2000) and care is taken with respect to choices of $N$ , $M$ and the domain truncation of $\unicode[STIX]{x1D701},\unicode[STIX]{x1D702}$ . To reduce the number of degrees of freedom, the mesh of nodes is non-uniformly spaced in both directions, to allow a higher resolution in the neighbourhood of $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D702}=0$ .

Figure 3.

Solutions to (1.9) obtained by parabolic marching in the downstream coordinate $x$ . A localised injection given by (3.3) is imposed with $x_{0}=3/2$ , $\unicode[STIX]{x1D6FE}=10$ , $c=1/10$ , with $\unicode[STIX]{x1D705}$ increasing in the direction of the arrows shown. (a) The centreline shear rescaled such that the undisturbed Blasius base flow has a value of $0.4696\ldots$ and $\unicode[STIX]{x1D705}=1,2,4,8$ . (b) A global measure of the streak amplitude defined by (3.5) for $\unicode[STIX]{x1D705}=2,4,8$ . The line segment shows the predicted decay ( $x^{\unicode[STIX]{x1D706}_{1}}$ ) obtained from the linear ( $\unicode[STIX]{x1D705}\ll 1$ ) eigenvalue analysis of § 2.

To represent the downstream developing solution we examine the centreline (dimensionless) shear at the wall:

We also present the global measure suggested by Andersson et al. (Reference Andersson, Brandt, Bottaro and Henningson2001)

where $\hat{U} _{B}$ is the Blasius solution ( $F^{\prime }$ in our earlier notation), so that $\hat{U} -\hat{U} _{B}=\tilde{U}$ in the notation of (1.8).

In figure 3(a) we show the evolution of the rescaled shear $\unicode[STIX]{x1D70F}(2x)^{1/2}Re^{-1/2}$ as a function of downstream distance $x$ , for an injection (3.3) with $\unicode[STIX]{x1D6FE}=10$ , $c=1/10$ , $x_{0}=3/2$ and increasing $\unicode[STIX]{x1D705}=1,2,4,8$ . It is worth noting that even for large-amplitude injection ( $\unicode[STIX]{x1D705}=8$ ) there is no evidence of any flow reversal and breakdown of the parabolic marching, instead the perturbation leaves a region of essentially zero shear until eventually starting to relax back towards the Blasius value of $0.4696\ldots \,$ . In the experimental work of Asai et al. (Reference Asai, Minagawa and Nishioka2002) it is also noted that there is no evidence of reverse flow behind the screen that provides the nonlinear disturbance mechanism.

Figure 4.

A measure of the maximum disturbance to the streamwise velocity as measured by ${\mathcal{A}}$ , defined by (3.5). The flow is perturbed nonlinearly via (3.1) with $\unicode[STIX]{x1D705}=8$ , $\unicode[STIX]{x1D6FE}=10$ . Four different cases are shown corresponding to (i) $x_{0}=3/2$ , $c=1/10$ , (ii) $x_{0}=5$ , $c=1/10$ , (iii) $x_{0}=10$ , $c=1/10$ and (iv) $x_{0}=10$ , $c=0.5$ .

Figure 5.

A solution of (1.9) subject to a perturbation of the form (3.1) (solid lines), compared to the (dimensional) experimental data (points) of Asai et al. (Reference Asai, Minagawa and Nishioka2002). The parameters in (3.3) are $\unicode[STIX]{x1D705}=12$ , $x_{0}=10$ , $\unicode[STIX]{x1D6FE}=10$ and $c=1/10$ ; this choice of $x_{0}$ corresponds to choosing $L^{\ast }=5~\text{cm}$ in the non-dimensionalisation.

Figure 3(b) shows the evolution of ${\mathcal{A}}$ for the same disturbance with $\unicode[STIX]{x1D705}=2,4,8$ , but over a substantially longer downstream length scale. The perturbation is long lived, with an ultimate decay that is algebraic; the figure also shows the predicted least-damped decay rate of $x^{-0.787}$ that arises from the (linearised) bi-global eigenmode prediction of $\tilde{U} \sim x^{\unicode[STIX]{x1D706}_{1}}$ (as per table 1).

The impact of the disturbance location and spanwise extent, as parameterised by $x_{0}$ and $c$ in (3.3), is shown in figure 4. Increasing $x_{0}$ decreases the amplitude of the resulting disturbance, being effectively over a length scale ( $1/\unicode[STIX]{x1D6FE}$ ) that is reduced compared to the local boundary-layer length scale (which is growing as $\sqrt{x_{0}}$ ). For larger values of $c$ the recovery towards a Blasius solution occurs over a shorter downstream distance. This is to be expected, as larger $c$ leads to a disturbance over a smaller spanwise length scale which in the nonlinear regime reduces more rapidly owing to spanwise diffusion. For more modest variations in the spanwise extent of the forcing, the streamwise centreline velocity of the streak is relatively insensitive. This is consistent with the experiments of Asai et al. (Reference Asai, Minagawa and Nishioka2002), which show only small-scale differences (at $z=0$ ) when comparing the streaks generated by screens of width 7.5 mm and 5.5 mm.

A more quantitative comparison can be made with the available experimental data. To do this we take the data of Asai et al. (Reference Asai, Minagawa and Nishioka2002) for a 7.5 mm mesh screen, comparing the reported velocity profiles at the centre of the streak ( $z=0$ ) with numerical results. Clearly we cannot precisely model the nonlinear disturbance associated with this choice of mesh screen, and we have not taken any systematic approach to maximise the level of agreement. Nevertheless, a comparison is shown in figure 5 for $x_{0}=10$ , $\unicode[STIX]{x1D705}=12$ with our default values of $\unicode[STIX]{x1D6FE}=10$ and $c=1/10$ . (Given the self-similar nature of the Blasius base-flow one could of course equally choose $x_{0}=1$ , $\unicode[STIX]{x1D6FE}=100$ , for example.)

Given that the mesh screen disturbs the flow at $x^{\ast }=500\,\text{mm}$ in the experimental configuration, this corresponds to a choice of $L^{\ast }=\,50\,\text{mm}$ in our non-dimensionalisation with $Re\approx 13\,540$ . The comparison of figure 5 is shown in terms of the dimensional coordinate normal to the plate, $y^{\ast }$ . The profile far from the disturbance source ( $z^{\ast }=30~\text{mm}$ ) is clearly in agreement with an undisturbed Blasius profile. In the wake of the screen, two profiles are presented corresponding to $x^{\ast }=550~\text{mm}$ ( $x=11$ ) and $x^{\ast }=700~\text{mm}$ ( $x=14$ ). This comparison provides some encouragement that (1.9) can effectively model the downstream response, however we make no claims beyond consistency of the results given that the computation is not precisely replicating the experimental disturbance of the mesh screen.

Figure 6.

Normalised experimental data (points) of Asai et al. (Reference Asai, Minagawa and Nishioka2002) mapped to the non-dimensional coordinate $\unicode[STIX]{x1D702}$ . Data points show the (normalised) deviation from a Blasius profile along the streak centreline ( $z=0$ , or equivalently $\unicode[STIX]{x1D701}=0$ ). Squares/circles correspond to data acquired at $x^{\ast }=550~\text{mm}$ and 700 mm respectively. The solid line is obtained from the least-damped ( $\unicode[STIX]{x1D706}_{1}$ ) eigenmode shown in figure 1(a).

To remove the ambiguity of the disturbance generation mechanism, we also compare these same experimental results with the bi-global eigenmodes discussed in § 2. To do this we subtract the (experimentally obtained) velocity profiles in figure 5 from the appropriate Blasius solution at the two locations $x^{\ast }=550$ , 700 mm, to give a nonlinear perturbation to the streamwise velocity along $z=0$ . After normalising the perturbation to have a unit maximum velocity, and converting the dimensional transverse coordinate $y^{\ast }$ to the dimensionless value $\unicode[STIX]{x1D702}$ , we compare these data with the corresponding centreline perturbation of the least-damped eigenmode shown in figure 1(a). Despite the obvious nonlinear nature of the perturbation at $x^{\ast }=700~\text{mm}$ (as shown in figure 5), the normalised response compares favourably with the shape of the first linear mode. In this regard it is worth noting that the second mode very closely reproduces the shape of the first mode along the centreline, whilst the third mode is of the same shape but displaced to slightly larger values of  $\unicode[STIX]{x1D702}$ .