2. Short blips

The typical tiny flexible blip is mounted on a solid surface such as an airfoil or bluff-body surface but the blip is so localised that the solid part of the surface may be considered as quasi-flat for most of our purposes. Concerning (first) the fluid dynamics, far from the surface the fluid is moving with a given constant velocity parallel to that flat part of the surface. The working below for two-dimensional flows incorporating short blips is expressed conveniently in terms of non-dimensional flow velocities $(u,v)$ , corresponding Cartesian coordinates $(x,y)$ , time $t$ and pressure $p$ , such that the dimensional versions are $U^{\ast }(u,v),L^{\ast }(x,y),L^{\ast }t/U^{\ast }$ and ${\it\rho}^{\ast }U^{\ast ^{2}}p$ , respectively. Here $U^{\ast }$ is the representative fluid velocity, taken to be the free-stream value above for definiteness, while $L^{\ast }$ is the typical length factor introduced in § 1, ${\it\rho}^{\ast }$ is the uniform density of the incompressible fluid and the temporal factor $L^{\ast }/U^{\ast }$ taken is the typical transport time. The continuity and Navier–Stokes equations of momentum balance are then

in vector form. The velocity vector $\boldsymbol{u}=(u,v)$ , pressure $p$ and coordinates $x,y$ are generally of order unity except near the flat part of the solid surface which is located along the axis $y=0$ . In particular $\boldsymbol{u}$ is given by $(1,0)$ in the far field and the leading edge of the airfoil can be taken as the origin without loss of generality. The Reynolds number is given by $\mathit{Re}=U^{\ast }L^{\ast }/{\it\nu}^{\ast }$ where ${\it\nu}^{\ast }$ is the uniform kinematic viscosity of the fluid. The blip starts at an order-unity distance downstream $x_{0}$ say from the airfoil leading edge, the representative length of the blip is now $\ell$ which is $\ell ^{\ast }/L^{\ast }$ and the prime concern is with the properties induced for short blips for which $\ell$ is small, when the Reynolds number is comparatively large.

For large values of the Reynolds number $\mathit{Re}$ a thin boundary layer is set-up along the undisturbed surface ahead of the blip and also downstream of it. The boundary layer is a classical one having $x$ scale of order unity whereas the $y$ scale is of order $\mathit{Re}^{-1/2}$ , with $u,p$ variations of $O(1)$ and $v$ being of $O(\mathit{Re}^{-1/2})$ . The time scale there is of order unity. This boundary layer and its local free stream form the oncoming and surrounding flow field for the blip-induced motion (see figure 1). The reasoning below based on estimates of the orders of magnitude present indicates that the main range of interest is for scaled lengths such that

The reasoning supposes the characteristic blip height $\mathit{Re}^{-1/2}{\it\delta}$ say to be significantly less than the boundary-layer thickness, which is of order $\mathit{Re}^{-1/2}$ since $x_{0}$ is of order unity (so ${\it\delta}$ is small), and supposes the major flow response to occur over the same height $O(\mathit{Re}^{-1/2}{\it\delta})$ in general by virtue of invoking a nonlinear response and the possibility of significant local alterations to the WSS for instance. The oncoming velocity profile then indicates that the typical velocity $u$ involved near the blip is small and comparable with ${\it\delta}$ because of the dominant wall-shear effect near the oncoming flat surface. The typical streamwise length scale $\ell _{1}$ of physical importance in the sublayer around the blip can then be estimated from balancing the order of magnitude of the inertial forces $uu_{x}\sim {\it\delta}^{2}/\ell _{1}$ against that of the prominent viscous force $\mathit{Re}^{-1}u_{yy}\sim \mathit{Re}^{-1}{\it\delta}/(\mathit{Re}^{-1/2}{\it\delta})^{2}$ , bearing in mind that $y$ is of scale $\mathit{Re}^{-1/2}{\it\delta}$ . The balance thus imposes $\ell _{1}$ as being of order ${\it\delta}^{3}$ . One would expect $\ell _{1}$ , $\ell$ to be comparable as the first central interactive case and so obtain a simple relationship between the fraction ${\it\delta}$ and the critical length scale with the accompanying scales

See Smith (Reference Smith1976a ,Reference Smith b ), Sobey (Reference Sobey1980), Lagrée (Reference Lagrée1994), Rothmayer & Smith (Reference Rothmayer, Smith and Johnson1998) and Lagrée (Reference Lagrée2007) for related discussions of scalings. Longer blips and shorter blips are mostly subcases: shorter ones for example produce the steepness effect that is addressed by Smith & Daniels (Reference Smith and Daniels1981) as regards breakaway separation.

Figure 1. The major parts of the flow/body interaction: the short-scale flow structure which comprises a viscous–inviscid sublayer surrounded by the main part of the incident boundary layer. Here (a) is for a single blip and shows the whole boundary layer while (b) is for the corresponding configuration with several or many blips, shown closer to the wall where to leading order the incident flow is a uniform shear.

So far we have taken the time scale to respond to the inertial force. In the next section we will take it to be slower, yielding quasi-steady behaviour; the time scale $l^{2/3}$ reasserts itself later however through a process of intensifications.

The lower limitation in (2.2a ) corresponds to the sublayer height $|y|$ becoming comparable with the streamwise scale $\ell$ and producing a quite tiny region governed by the full system (2.1) in normalised form. In contrast the upper limitation in (2.2a ) is associated with the triple deck stage where the thin sublayer around the blip experiences a substantial feedback of pressure which arises from interaction with the flow outside the surrounding boundary layer. In between, where the range (2.2a ) applies, the sublayer is controlled by thin-layer dynamics alone. Moreover, the range of validity in (2.2a ) which is verified above is actually quite a large one in terms of the scales covered.

The flow structure is therefore concentrated primarily in the thin sublayer of figure 1. The flexible blip of unknown shape in 1(a) occupies the range $0<X<1$ and has height comparable with the sublayer height, whereas the surface is fixed and solid outside that range; 1(b) is for several or many blips. The blips can be humps or dents. The thick dashed line between the sublayer and the rest of the boundary layer indicates the lack of displacement over these short length scales. In the sublayer at leading order

and all of the capital-lettered quantities are generally of order unity. The full system (2.1) then reduces to the well-known condensed flow interaction (Smith Reference Smith1976a ,Reference Smith b ; Smith & Daniels Reference Smith and Daniels1981) given by

with the unknown scaled pressure $P(X,T)$ being independent of $Y$ because of the $y$ -momentum equation. The relevant boundary conditions are

for no slip on the moving blip surface and to match with the outer flow response in turn. The requirement (2.4f ) of effectively zero displacement in $Y$ corresponds to the feedback effect from the flow outside the sublayer being relatively small. The function $f$ in (2.4d,e ) denotes the unknown scaled shape of the blip surface which is addressed further just below while the positive $O(1)$ factor ${\it\lambda}$ in (2.4f ) is the given scaled incident WSS, namely $\mathit{Re}^{-1/2}(\partial u/\partial y)$ at $y=0$ , in the surrounding boundary layer locally: see figure 1.

Concerning (second) the blip surface shape as it responds to the fluid flow over the blip, the shape and the flow interact via the local pressure as in the models used by Carpenter & Garrad (Reference Carpenter and Garrad1985), Gajjar & Sibanda (Reference Gajjar and Sibanda1996), Davies & Carpenter (Reference Davies and Carpenter1997) and Pruessner (Reference Pruessner2013) and others. The assumptions made are primarily those of the widely used membrane-model type as in the references immediately above with particularly interesting background discussions of linearly elastic materials and allied facets relevant here being in Takagi & Balmforth (Reference Takagi and Balmforth2011) as well as Vella, Kim & Mahadevan (Reference Vella, Kim and Mahadevan2004), Stewart, Waters & Jensen (Reference Stewart, Waters and Jensen2009), Singh, Lister & Vella (Reference Singh, Lister and Vella2014) and Xu, Billingham & Jensen (Reference Xu, Billingham and Jensen2014). This then gives the wall equation

In this simple plate membrane model the unknown blip shape is $y={\it\eta}(x,t)$ and the non-dimensional constant coefficients $e_{n}$ are $(e_{1},e_{2},e_{3},e_{4},e_{5})=(-B^{\ast }/U^{\ast 2}L^{\ast 3},T_{t}^{\ast }/U^{\ast ^{2}}L^{\ast },-{\it\kappa}^{\ast }L^{\ast }/U^{\ast 2},M^{\ast }/L^{\ast },C^{\ast }/U^{\ast })/{\it\rho}^{\ast }$ with $M^{\ast },C^{\ast },B^{\ast },{\it\kappa}^{\ast },T_{t}^{\ast }$ being the mass density, the damping constant, the flexural rigidity, the spring stiffness and the longitudinal tension respectively, while $p_{0}$ is the non-dimensional base pressure relative to the oncoming pressure level which is taken to be zero. The values of the above quantities are determined experimentally or are tabulated for certain materials. The scaled version appropriate to the short-blip application is then

where ${\it\eta}=\mathit{Re}^{-1/2}{\it\delta}f$ , $\tilde{e}_{1}=\mathit{Re}^{-1/2}e_{1}/(\ell ^{4}{\it\delta})$ , $\tilde{e}_{2}=\mathit{Re}^{-1/2}e_{2}/(\ell ^{2}{\it\delta})$ , $\tilde{e}_{3}=\mathit{Re}^{-1/2}e_{3}/{\it\delta}$ , $\tilde{e}_{4}=\mathit{Re}^{-1/2}e_{4}/{\it\delta}^{5}$ , $\tilde{e}_{5}=\mathit{Re}^{-1/2}e_{5}/{\it\delta}^{3}$ , $P_{0}=\ell ^{-2/3}p_{0}$ . In particular $\tilde{e}_{1}<0$ , $\tilde{e}_{2}>0$ , $\tilde{e}_{3}<0$ . The coefficients are sensitive to the blip length scale $\ell$ (recall ${\it\delta}$ in (2.2b−f )) and to $\mathit{Re}$ but in applications of interest here the dominant coefficient can sometimes be $\tilde{e}_{2}$ . The boundary conditions on $f$ are

if the end points of the single blip are taken as $X=0,1$ for definiteness. Representative initial conditions have the shape $f$ and pressures $P,P_{0}$ being zero.

The nonlinear governing system is therefore (2.4a–f ), (2.6a,b ). Fair agreement between theoretical results based on the system and experimental results or direct simulations is found in the prescribed-shape case (Smith Reference Smith1976a ,Reference Smith b ; Sobey Reference Sobey1980; Rothmayer & Smith Reference Rothmayer, Smith and Johnson1998). This system whether with prescribed wall shape or flexible shape then appears to require numerical treatment usually: see the next section. On the other hand if $P_{0}$ is zero then there is simply no disturbance. Thus, a linearised version which is also helpful and intriguing in its own right has $f$ being small and so (2.4a–f ) reduce to linear equations for small $U-{\it\lambda}Y,V,P$ . Use of a Fourier transform in $X$ for example as in Stewartson (Reference Stewartson1970), Smith (Reference Smith1976a ,Reference Smith b ) and Guneratne & Pedley (Reference Guneratne and Pedley2006) along with inversion and convolution properties then yields for quasi-steady flow the integro-differential relation

between $P,f$ for $X>0$ , thus replacing (2.4a–f ). Here ${\it\gamma}=-3\,\text{Ai}^{\prime }(0){\it\lambda}^{5/3}/{\it\Gamma}(1/3)$ is a positive constant $\{{\approx}0.289838{\it\lambda}^{5/3}\}$ . The parabolic nature of the flow contribution on its own is then clear. Also the unknown blip $f$ is again assumed to start at the station $X=0$ for definiteness and the pressure $P$ there is zero. In the streamwise direction the only boundary conditions imposed on $U,V,P$ are at the upstream end of the first blip, where $U={\it\lambda}Y$ , $V=P=0$ , and we note that it is inappropriate to fix the downstream pressure on the current short length scales. The nature of the problem is somewhat mixed, being parabolic upstream of the blip(s), then elliptic over the length of a blip, then parabolic at all streamwise stations where there is no blip, and so on, but the overall upstream influence of a relatively long triple-deck in external flow (Stewartson Reference Stewartson1970; Smith Reference Smith1982) or the long $\mathit{Re}^{1/7}$ (multiplied by the channel width) axial length in internal flow (Smith Reference Smith1977; Luo & Pedley Reference Luo and Pedley1996) is insignificant for the present short blips. Likewise the corresponding flow-dominated linear instabilities existing over the longer scales are inactive in the present setting. The fully nonlinear unsteady condensed flow (2.4a )–(2.6b ) with $|\partial /\partial T|$ of order unity can nevertheless lead to a finite-time breakup, Smith (Reference Smith1988) and Peridier, Smith & Walker (Reference Peridier, Smith and Walker1991), as a nonlinear by-pass into deep transition as seen later in §§ 4 and 6. The motivational settings of § 1 however are more concerned with quasi-steady behaviour initially in which $|\partial /\partial T|$ is relatively small.

It can be seen immediately that a positive $f$ shape corresponding to a bump rather than a dent provokes negative pressures, which makes sense physically in terms of an adverse pressure gradient on the front face of the blip and a favourable pressure gradient most likely on the rear face, together with a lag in the response due to convolution. Likewise a pressure rise followed by a fall is expected for a dent. The persistence of the pressure response downstream of the end point $X=1$ is evident from (2.7) (the integral is then from 0 to 1) and represents a wake effect. A normalisation is now applied. With $U,V,P,Y,T,f$ normalised by factors ${\it\lambda}^{m}$ where $m$ is $2/3,1/3,4/3,-(1/3),-(2/3),-(1/3)$ in turn (see the references above) and the representative $X$ already normalised to unity by definition of blip length, the flow contributions throughout (2.4) are thus normalised to $O(1)$ whereas the shape contributions in (2.6a ) have orders $|\tilde{e}|{\it\lambda}^{-1/3}$ on the left compared with ${\it\lambda}^{4/3}$ and $P_{0}$ successively on the right-hand side. Here only the steady case is considered for now and $|\tilde{e}|$ refers to the typical size of the wall coefficients $\tilde{e}_{n},~n=1,2,3$ , all of the wall coefficients being supposed to be of the same order for convenience. Hence the active ratios indicate two major parameters, namely ${\it\Gamma}_{1},{\it\Gamma}_{2}$ defined by

When ${\it\Gamma}_{1},{\it\Gamma}_{2}$ are of order unity the fluid-shape interaction is fully nonlinear, while if ${\it\Gamma}_{1}$ is small then linearised theory holds since the normalised responses in $P,f$ become of order ${\it\Gamma}_{1}$ . In the subsequent calculations we tend to take $|\tilde{e}|$ as given as $O(1)$ and vary the incoming shear factor ${\it\lambda}$ and the relative base pressure $P_{0}$ in order to help explore the main two-dimensional parameter space. There are many other parameters of course detailing relative wall coefficients and blip distributions for example. The linear and nonlinear responses for a single blip are examined in the next two sections whereas those from the presence of multiple blips are addressed in § 5.

3. Results and intensifications

The influences of various physical factors are discussed in the following subsections. Analytical and numerical treatments of (2.4a–f ), (2.6a,b ) appear necessary. The numerical method we adopted is a finite difference one and, in brief, it comprises an iterative process of guessing the scaled shape $f$ for all $X$ in the range, then solving (2.4a–f ) for the scaled pressure $P$ or using (2.7) if applicable, then updating $f$ from integration of (2.6a,b ), and continuing to iterate thus between the flow and wall equations until successive iterates differ by a specified small tolerance, typically 10⁻⁶ in $P$ . This is with a prescribed under-pressure value $P_{0}$ in (2.6a ) and with a Prandtl transposition introduced into (2.4a–f ) in order to deal with the no-slip requirement accurately. The treatment applies with modifications for both the linear and the nonlinear cases, abetted by analysis in §§ 3–5, for quasi-steady slowly varying flow. Results are presented in figures 2–9. These tend to confirm the expectation from the previous section of adverse and favourable pressure gradients over the typical blip as well as corresponding reductions and increases respectively in the scaled WSS, $\partial U/\partial Y$ at $Y=0$ , along with response lags and with a clear wake effect in terms of the wall pressure downstream of the blip. Both linear and nonlinear quasi-steady flows will be considered at first below before the emphasis moves on to the linear regime for reasons (namely the occurrence of intensifications and blowups) that will become apparent.

Figure 2. (a) Numerical results for a single blip with wall coefficients $[\tilde{e}_{1},\tilde{e}_{2},\tilde{e}_{3}]$ of $[-1,1,-1]$ , incoming shear ${\it\lambda}$ of unity and $P_{0}=1$ or $-1$ , showing blip shape $f$ , slope $f^{\prime }$ and fluid-flow pressure $P$ versus scaled distance $X$ . Linearised and low-amplitude nonlinear calculations are compared (light and dark). (b) As (a) but ${\it\lambda}$ varying from 1 to 80: linearised results for shape. (c) Normalised shape at ${\it\lambda}={\it\lambda}_{c}\approx 58.504$ , the first intensification value, from linearised theory.

4. Weakly nonlinear behaviour of a single blip near intensification

To clarify how nonlinearity first affects intensification at low amplitude this section considers the weakly nonlinear response of the blip shape and the local flow properties at parameter values close to a typical intensification. Near a representative critical value ${\it\lambda}_{0}$ for intensification the scaled incident WSS can be expressed as

say, where the constant ${\it\epsilon}$ is small and the coefficients ${\it\lambda}_{0},{\it\lambda}_{1}$ and so on are of order unity. The entire fluid-body interaction then takes the form

for $M=1,2,3$ , when nonlinear influences first become substantial. The blip-shape perturbation of order ${\it\epsilon}$ is an order of magnitude greater than the base-pressure perturbation (of order ${\it\epsilon}^{2}$ ) in (4.2b ), hence providing evidence of the occurrence of intensification subject to the self-consistency that emerges below. The critical disturbance function $f_{1}$ is taken to be normalised such that $f_{1}^{\prime \prime }(0)=1$ , for the sake of definiteness, whereas $A$ represents the unknown disturbance amplitude and is dependent on the slow scaled time $\tilde{T}$ in general which is defined by time $T={\it\epsilon}^{-1}\tilde{T}$ . The physical expectation is that the slow-time dependence is present usually because of variations in the under-pressure or in the change in the wall-shear factor ${\it\lambda}_{1}$ , while the leading factor ${\it\lambda}_{0}$ must normally remain constant for physical sense in the scalings. If ${\it\lambda}$ is evolving faster, on the condensed scale with $T$ of $O(1)$ , then the far-field conditions involve a Stokes layer (Brown & Daniels Reference Brown and Daniels1975) wherein $Y$ is $O(1)$ balancing the order of magnitude $T^{1/2}$ from (2.4d,e ) at large $|X|$ . For slow evolutions however the effective time scale is larger and so is the Stokes $Y$ scale, even lying outside the condensed layer of § 2 if the effective time scale is as large as a positive power of $\mathit{Re}$ . Further the evolution in (4.1)–(4.2c ) is with regard to the perturbation in ${\it\lambda}$ rather than the leading-order contribution. A weak temporal dependence through the additional wall-flexibility coefficients $\tilde{e}_{4},\tilde{e}_{5}$ in (2.6a ) would also be admissible and is discussed later in § 6 but we focus here on temporal dependence springing from the fluid-dynamical background. The relative smallness of the under-pressure in (4.2b ) is notable, its order- ${\it\epsilon}^{2}$ size being due to the interaction intensifying at the leading order ${\it\epsilon}$ . Further the contributions to the wall-flexibility parameters of order ${\it\epsilon}$ and smaller on the right-hand side of (4.2c ) correspond to detuning of the intensification; a similar effect can be generated by slight detuning of the blip length.

Substitution of (4.1)–(4.2c ) into the nonlinear governing equations (2.4a–f ), (2.6a,b ) along with the usual stream-function properties yields at leading order $O({\it\epsilon})$ the unforced linear system

in which $m$ is unity and $R_{11}$ – $R_{31}$ are identically zero. Here $X$ lies between $0,1$ , the values of $f_{m},f_{m}^{\prime }$ are to be zero at the end points and the intensification or eigen-function which is accentuated through the absence of any forcing on the $m=1$ quantities is described below. At next order, $O({\it\epsilon}^{2})$ , the same system applies but with $m=2$ and with the forcing that arises from detuning, evolutionary and nonlinear contributions, so thatThese forcings are affected by the leading-order flow and wall solutions of course as well as by the small under-pressure. In particular $R_{12},R_{22}$ are flow-field effects while $R_{32}$ constitutes the wall effect.

The solution of (4.3a–e ) may be obtained by means of a Fourier transform (superscripted $F$ ) applied in $X$ , which allows (4.3a,b ) to be cast as a forced Airy equation for the scaled flow shear. Hence,

where ${\it\alpha}$ is the transform variable, $\text{Ai}$ is the Airy function, ${\it\xi}=(\text{i}{\it\alpha}{\it\lambda}_{0})^{1/3}Y$ and

Use of (4.3c,d ) then enables the unknown shear constant $B_{m}$ to be determined in terms of the scaled pressure which is found to be given by

Here

So the wall balance (4.3e ) requires the relation

to hold, in which $R_{42}(X)={\it\Gamma}(X)+R_{32}(X)$ . The factor multiplying the integral on the right in (4.7a ) agrees with the intensified value of that in (2.7) as required for consistency. The relation (4.7a ), which is subject to the end conditions

acts at leading order $(m=1)$ to define the intensified blip shape in normalised form but also acts, at the second order $(m=2)$ where the $R$ terms are non-zero, to provide a solvability constraint that helps control the unknown amplitude. This constraint stems from use of the normalised function $q(X)$ say that is adjoint to $f_{1}(X)$ and can be shown from (4.7a,b ) when $m=1$ to be given by $q(X)=f_{1}(1-X)$ . Multiplication through by $q(X)$ in (4.7a ) when $m=2$ then leads to the requirement that

which constitutes an amplitude equation. A substitution of the effects from (4.4a–c ) in the requirement (4.8) yields the amplitude equation for $A(\tilde{T})$ as

where

and it can be shown that all the coefficients $d_{1}{-}d_{6}$ are positive; a calculation gives the estimates $0.422,0.392,14.76,0.362,0.029,1.11$ in turn. The latter values are for the central example of the wall coefficients $\tilde{e}_{1},\tilde{e}_{2},\tilde{e}_{3}$ being $-1,1,-1$ respectively whereas effects over a wide range of variation in these wall coefficients are investigated in appendix A.

Figure 10. (a) The phase-plane solution with horizontal axis $A^{\ast }=d_{6}A/|d_{7}|$ and vertical $\stackrel{\bullet }{A}^{\ast }$ for times $|d_{7}|\tilde{T}/d_{1}$ , associated with weakly nonlinear evolution in or near intensification. (b) Evolution of amplitude $A$ when $d_{7}=\tilde{T},P_{02}=0.25$ with $d_{1},d_{6}$ normalised to unity, for various small kicks in amplitude at large negative $\tilde{T}$ . Quasi-steady states denoted $\mathit{qss}$ are shown for comparison. (c) As (b) but $P_{02}=-0.25$ .

The amplitude equation (4.9a ) encapsulating the nonlinear response in terms of the amplitude function $A(\tilde{T})$ is Riccati-like and yields the phase-plane diagram drawn in figure 10(a), when $d_{1},d_{6},d_{7}$ are independent of time $\tilde{T}$ , which tends to confirm the significance of only the signs of the coefficients above. Varying the scaled pressure $P_{02}$ yields the different sets of trajectories with positive, zero or negative intercept values ( $+,0,-$ respectively) at zero $A$ while $d_{7}$ being positive or negative is responsible for the left–right reflection of the curves. Here $d_{7}$ may be positive or negative depending on the four parts which constitute that coefficient. The arrows indicate direction along the trajectories with increasing time: rightwards in the upper half plane and leftwards in the lower half. The normalised diagram, plotting $(d_{6}d_{1}/|d_{7}|^{2})\,\text{d}A/\text{d}\tilde{T}$ against $d_{6}A/|d_{7}|$ , shows first that if the scaled under-pressure $P_{02}$ is negative then on any solution trajectory there are two roots of the quadratic equation corresponding to zero time derivative, one root having positive amplitude and the other negative, and all trajectories are also connected with a large-amplitude asymptote in which $\text{d}A/\text{d}\tilde{T}\propto A^{2}$ with a negative coefficient of proportionality. On the other hand if $P_{02}$ is positive then two roots are found if ${d_{7}}^{2}>4d_{6}P_{02}$ (in which case the roots are of the same sign), one root in the marginal case where equality holds and no real roots otherwise. Again the connection with a large-amplitude asymptote is clear. Whether the under-pressure is positive or negative, the greater of the two amplitude roots acts as an attractor and the lesser one as a repellor whenever two roots exist, whereas the large-amplitude asymptote which is featured on the left-hand side of the diagram with $A$ tending to negative infinity serves as an attractor in every instance. The initial value of $A$ and subsequent evolution with scaled time can play a key role in deciding the eventual outcome for the evolving amplitude. The choice of outcomes is

The steady-state value in (4.10a ) can be positive or negative, being dependent upon the values of $P_{02}$ and the coefficient $d_{7}$ , and is consistent with the value $-P_{02}/(d_{2}{\it\lambda}_{1})$ for the linear steady regime and results of § 3 when ${\it\lambda}$ approaches the critical value identified there with prescribed $\tilde{e}_{n}$ coefficients. This links with the quasi-steady behaviour of § 3 at low amplitudes.

In contrast the blow-up behaviour in (4.10b ) occurs at a finite scaled time and is such that the fully nonlinear system (2.4a–f ), (2.6a,b ) is reinstated over a shorter time scale around the blow-up time $T$ of approximately ${\it\epsilon}^{-1}\tilde{T}_{0}$ . This is because when $T-{\it\epsilon}^{-1}\tilde{T}_{0}$ becomes of order unity the denominator in (4.10b ) becomes of order ${\it\epsilon}$ , making $A$ grow as large as ${\it\epsilon}^{-1}$ and therefore causing the $U$ -perturbation and $|f|$ in (4.2a,b ) to become of order unity. Saturation as in (4.10a ) and blow-up to full (strong) nonlinearity within a finite scaled time as in (4.10b ) are the two main possibilities for the weakly nonlinear near-intense response, then, while the latter full nonlinearity leads on to the strong break-up route discussed later in terms of transition.

Concerning (4.9a,b ) further the results in figure 10(b,c) are formally for varying ${\it\lambda}_{1}$ from large negative to large positive values which force $d_{7}$ to pass smoothly from negative to positive; specifically this is normalised to $d_{7}=\tilde{T}$ with $d_{1}=d_{6}=1$ . The scaled base pressure is $\pm 0.25$ . The low-amplitude asymptotes for large negative times $\tilde{T}$ match with the quasi-steady behaviour found in § 3 whereas blow-up of the amplitude $A$ can be produced within a finite time, confirming the nonlinear progression to the strong break-up route mentioned at the conclusion of the previous paragraph.

5. Intensifications for many blips

Interactive solutions for multi-blips are now considered. Solutions for four blips are presented in figure 11(a,b), with the values of the wall coefficients fixed at $(-1,1,-1)$ . These are linear results. Also solutions for one, two or three blips correspond to terminating the calculation just before the second, third or fourth blip successively. In figure 11(a) two different arrangements of the four under-pressures are considered, for parameter ${\it\lambda}$ of unity, and it is apparent that the interactions between the successive blips are relatively mild in these cases. In figure 11(b) ${\it\lambda}$ is increased to a near-critical value of 55 where the interactions are seen to be considerable. When $P_{0}$ is positive the minimum pressure produced decreases with downstream distance $X$ and the maximum pressure also decreases, and in addition the scaled pressure $P$ starts from a non-zero value at the beginning of all but the first blip because of the wake effect. For linear or nonlinear intensifications the wake effects in the induced pressure field can play a substantial role in the total interaction produced by a configuration of many blips, as described below.

Figure 11. Two or more blips. (a) Linearised $f,P$ versus $X$ with successive base-pressures $P_{0}$ of $1,1,1,1$ (thin curves) or $1,-1,1,-1$ (thick), blip lengths unity, ${\it\lambda}=1$ , wall case $[-1,1,-1]$ . (b) Linearised $f,P$ with base pressures $1,1,1,1$ , blip lengths unity, ${\it\lambda}=55$ (just below intensification), wall case $[-1,1,-1]$ . (c) Weakly nonlinear heights $f$ for the equimaterial setting of (5.1), equally or unequally spaced blips, yielding heights ${\it\epsilon},{\it\epsilon}^{1/2}$ , etc., as $X$ increases; not to scale. Results for 2 blips stop before the third, 3 blips stop before the fourth, and so on.

Figure 12. Sketch, not to scale, of effects of changing incident shear  ${\it\lambda}$ : curve $a$ shows response of max $|WSS|$ if ${\it\lambda}$ varies slowly as in $b$ from subcritical to supercritical; when ${\it\lambda}$ variation is instead as in $c$ the response remains close to  $c$ . Time scales (i)–(iv),… are slow, fast, faster, faster, ….

Supposing the configuration consists of a sequence of blips that are of identical length (normalised to unity), identical material and so on, but not necessarily spaced out equally, we consider the most upstream one first. This approach ties in with the mixed parabolic–elliptic nature of the entire fluid/body interaction. Near or in intensifications the first blip produces a dominant scaled pressure response $P$ and scaled shape response $F$ which in the incipient nonlinear range have amplitude of order ${\it\epsilon}$ , that is small as in § 4, and these act as eigenfunctions. The associated under-pressure $P_{0}$ is only of order ${\it\epsilon}^{2}$ then and acts at second order to help control the detailed amplitude evolution; thus, the dominant pressure response varies as the square root of the forcing pressure. The main point now concerns the $O({\it\epsilon})$ pressure $P$ and the fact that it exhibits a wake contribution as referred to in § 3, a contribution which extends over the second blip in particular. As a consequence the forcing pressure for the second blip is also of $O({\it\epsilon})$ . Moreover this blip is augmented, as it is identical to the first blip, and so it produces the very same eigenfunction response as before except that, crucially, the amplitude is of order ${\it\epsilon}^{1/2}$ since the forcing pressure (replacing under-pressure effect) is now of order ${\it\epsilon}$ . The overall effect is therefore larger on the second blip and begins as if anew. Also the mechanism can continue afresh to further blips downstream: the pressure wake effect from the second blip is $O({\it\epsilon}^{1/2})$ , extends over the third blip, produces $O({\it\epsilon}^{1/4})$ amplitudes in pressure and shape on the third blip; the pressure wake effect from the third blip is $O({\it\epsilon}^{1/4})$ , extends over the fourth blip, produces $O({\it\epsilon}^{1/8})$ amplitudes in pressure and shape there; and so on.

The prediction then is of an increasing trend of pressure and shape amplitudes with increasing distance downstream, and likewise for the wall-shear perturbations. The trend has the enhanced amplitude sequence:

from blip to successive blip for the above case of several identically constituted blips (see figure 11 c) whereas for example the sequence is

for the case of two identical blips followed by any two differently constituted blips and then identical ones again. Each of the two different blips here provokes a maintained amplitude in shape and pressure because of the wake-pressure effect from its immediate predecessor and so delays the enhancement but does not prevent it continuing downstream. Overall, the increasing downstream trend and mechanism involved seem to add weight to the significance of non-periodicity in the overall configuration.

The same or similar reasoning extends to configurations other than the identical-blips one above, as (5.2) indicates. A combination of the workings in this and the previous sections points to a succession of linked nonlinear amplitude equations which are of considerable interest for future research.

6. Further comments

Three points (i)–(iii) stand out in conclusion.

(i) Inferences on flow transition. The work has shown that when the wall shear of the incident boundary layer flow acquires certain critical values (eigenvalues) a tiny flexible blip can, after some delay, rise or fall to a magnitude much greater than is otherwise the case. This intensification (§ 3, § 5) along with blowup (§ 4 and see next paragraph) points to a tuneable trip being feasible over a wide parameter range in the boundary layer. Both shear and wall flexibility are required for the present intensifications and fast growth mechanisms to occur as the present scales lie outside the usual boundary-layer instability of Tollmien–Schlichting waves (Smith Reference Smith1979).

Intensification leads to a non-standard path into transition from low amplitudes, as figure 12 indicates by showing schematically the effects of gradually changing the oncoming shear ${\it\lambda}$ for the regime of relatively low base pressures. Curve $a$ shows the response of the maximal $|WSS|$ when ${\it\lambda}$ varies slowly as in $b$ from subcritical to supercritical. The response produces only a small disturbance from ${\it\lambda}$ over the quasi-steady temporal scale (i) of § 3 until a critical value ${\it\lambda}={\it\lambda}_{c}$ is encountered at which stage (ii) weakly nonlinear amplification can occur (§ 4), leading to strongly nonlinear evolution over the faster time scale (iii) of (2.4a–f ), (2.6a,b ). This is followed by finite-time blowup as in Smith (Reference Smith1988) and Peridier et al. (Reference Peridier, Smith and Walker1991) which provokes the even faster evolution (iv) described by Davies, Bowles & Smith (Reference Davies, Bowles and Smith2003) (see also Cassel & Conlisk Reference Cassel and Conlisk2014; Gargano et al. Reference Gargano, Sammartino, Sciacca and Cassel2014) with further restructuring and deep transition towards turbulence taking place. The time scales (i)–(iv) etc. for response $a$ are slow, fast, faster, faster, etc. The boosted nonlinear behaviour in $a$ does not arise however without the intensification associated with a critical ${\it\lambda}$ . Thus when the incident ${\it\lambda}$ variation is subcritical throughout as in $c$ the response remains close to $c$ . It is noted that the incident ${\it\lambda}$ values $b,c$ do not change scale. The variation in surrounding conditions can be by means of varying Reynolds number rather than shear as in the figure.

(ii) Comparisons. A connection can be made here with the recent work on non-flexible but dynamic roughnesses (Huebsch Reference Huebsch2006; Huebsch et al. Reference Huebsch, Gall, Hamburg and Rothmayer2012) which are similarly small and numerous distributed on an airfoil surface. These show considerable effects on transition and on the mean-flow alteration in particular similar to those discussed above. Detailed comparisons may follow. An additional contrast can be made with the works on internal motions cited in the introduction, since the present study based on small-scale condensed flow holds also for internal motions with comparatively short flexible blips. In particular Guneratne & Pedley (Reference Guneratne and Pedley2006), Kudenatti et al. (Reference Kudenatti, Bujurke and Pedley2012) and Pihler-Puzović & Pedley (Reference Pihler-Puzović and Pedley2013) and related papers are all on longer length and time scales and largely involve the $1/7$ -power length scaling of viscous–inviscid interaction that brings cross-channel pressure gradients into play (Smith Reference Smith1977) as well as upstream influence in the steady case and Tollmien–Schlichting-like instabilities in the unsteady case (Hall & Smith Reference Hall and Smith1982). Such overall upstream influence and underlying flow instabilities are absent in effect over the short scales of current concern whether the flow is external or internal. Moreover, the current concern includes influences (due to $\tilde{e}_{1},\tilde{e}_{3}$ ) from flexural rigidity and spring stiffness which are absent in the papers above. Nevertheless comparison is possible. The results in our figures 7, 9(c) with regard to higher roots can be compared, with rigidity and stiffness neglected, and they seem to agree with those in Guneratne & Pedley’s (Reference Guneratne and Pedley2006) figure 5 in two ways: the first is for ‘shorter’ length scales (large ${\it\sigma}$ in Guneratne and Pedley) and the second is for ‘longer’ scales (small ${\it\sigma}$ ). Both of these ranges of length scales yield fixed displacement of the wall layer(s) in effect, namely zero or the average of the wall shapes respectively as described in Smith (Reference Smith1976a ,Reference Smith b ), Smith (Reference Smith1977) and Sobey (Reference Sobey2000), the former length scales being analogous with our range (2.2a ). The comparison takes into account the different incident wall shears and wall coefficients for the internal and external applications and the factor of two associated with the average mentioned above in the case of much longer length scales. The present theory thus covers a wide range in internal channel flows, specifically

with allowance for that factor of two (and noting that this $\mathit{Re}$ is based on channel width). The intensified or critical values from (3.4a,b ) for short blips are in the same close neighbourhood as the numerical values in Guneratne & Pedley (Reference Guneratne and Pedley2006) at large ${\it\sigma}$ . The corresponding analytical values for ‘long’ blips are, from the factor of two above, simply given by

The change in critical values here is due to the mitigating effect of the opposite channel wall for the ‘long’ cases. In comparison, on a logarithmic scale the numerical values of spacings between the large- ${\it\sigma}$ roots are approximately the same as those between small- ${\it\sigma}$ roots in Guneratne & Pedley (Reference Guneratne and Pedley2006), suggesting consistency with the prediction (6.2).

(iii) Further work. It will also be interesting to extend the tiny-flexible-blip study to the highly nonlinear regime as in Smith & Daniels (Reference Smith and Daniels1981) producing longer-scale separations. Some evidence for this appears in Smith (Reference Smith1976a ,Reference Smith b ) and Pihler-Puzović & Pedley (Reference Pihler-Puzović and Pedley2013). In terms of the parameters of § 2 the regime corresponds to ${\it\Gamma}_{1}\sim 1$ but ${\it\Gamma}_{2}\gg 1$ . Future studies should investigate unsteadiness associated with the influences of mass density and damping in the wall response in addition to that considered in § 4. The initial effect of the damping proportional to $e_{4}$ in (2.5) is merely to alter the coefficient of the single derivative in (4.9a ), raising the interesting possibility of a change in sign of the coefficient and hence a change in the physical mechanism, whereas that of the mass density represented by $e_{5}$ is to introduce a second derivative in scaled time and this alters the physical nature and implications of the amplitude equation to a greater extent.