## Abstract

Interactive solutions for steady two-dimensional laminar marginally separated boundary layers are known to exist up to a critical value *Γ*_{c} of the controlling parameter (e.g. the angle of attack of a slender airfoil) *Γ* only. Here, we investigate three-dimensional unsteady perturbations of such boundary layers, assuming that the basic flow is almost critical, i.e. in the limit *Γ*_{c}−*Γ*→0. It is then shown that the interactive equations governing such perturbations simplify significantly, allowing, among others, a systematic study of the blow-up phenomenon observed in earlier investigations and the optimization of devices used in boundary‐layer control.

## 1. Introduction and motivation

As a starting point, we consider the flow of an incompressible fluid past a slender airfoil at a small angle of attack *α* (figure 1). Also let us assume that the Reynolds number formed with the free-stream velocity , the chord length of the airfoil and the kinematic viscosity is large, , so that viscous effects are confined to a thin boundary layer near the solid wall where the no slip condition has to be satisfied. It is then well known that solutions of the classical boundary‐layer equations for laminar flow exist up to a critical value *α*_{c} only where the wall shear stress distribution *τ*_{w}(*x*) exhibits the marginal separation singularity, i.e. *τ*_{w} vanishes in a single point but immediately recovers and the displacement thickness *δ**(*x*) shows a kink (figure 2). In reality, a short separation bubble forms eventually if *α* is further increased, and this phenomenon can still be studied by means of the boundary-layer equations, provided that the pressure gradient is no longer taken to be imposed but allowed to adjust owing to the local interaction between viscous and inviscid flow regions. By applying such an interaction strategy, one obtains the integro-differential equation(1.1)for the scaled wall shear stress *A*∝*τ*_{w}(*X*, *Z*, *T* ), which contains the single controlling parameter *Γ*∝(*α*−*α*_{c}) while *λ*, *γ* and *κ* are positive constants (Braun & Kluwick 2004). The flow is taken to consist of a two-dimensional steady state and unsteady three-dimensional perturbations: *A*=*A*_{∞}(*X* )+*A*_{1}(*X*, *Z*, *T* ) Here, *X*, *Z* and *T* denote, respectively, the local distances in the streamwise direction, the lateral direction and the time, and all quantities are suitably non-dimensionalized. Also taken into account are the effects of so-called smart devices used in boundary‐layer control, such as surface‐mounted obstacles with local height *h*=*h*_{∞}(*X* )+*h*_{1}(*X*, *Z*, *T* ) and/or suction strips with local suction velocity *v*_{w}=*v*_{w∞}(*X* )+*v*_{w1}(*X*, *Z*, *T* ) (see insert of figure 1). Steady two-dimensional uncontrolled marginally separated flows were studied first by Ruban (1981) and Stewartson *et al*. (1982). Important features of such flows can be derived from figure 3*a* in which the value of the wall shear at *X*=0 is plotted as a function of *Γ*. It is seen (i) that interactive solutions exist up to a critical value *Γ*_{c} of *Γ* and (ii) that these solutions are non-unique for *Γ*>0, where they form an upper and lower branch. To shed further light on this flow behaviour, Navier–Stokes solutions have recently been calculated by Braun *et al*. (2003) for the airfoil shown in figure 1 using the DLR-TAU-code (e.g. Gerhold *et al*. 1997), where the Spalart–Allmaras turbulence model is implemented, which allows one to specify the location of transition. Converged solutions could be obtained up to a maximum angle of attack *α*_{m}=4.25°, which differs only slightly from the prediction *α*_{m}=4.44° resulting from the theory of marginal separation based on *Γ*_{c}=2.66 and the value *Re*=2×10^{4} used in the numerical computations (figure 6*b*). Even more interestingly, laminar–turbulent transition had to be allowed for to achieve convergence, and for *α*≈*α*_{m}, transition was fixed to occur approximately at the location of the laminar separation bubble. Support for this observation appears to be provided by recent studies of Alam & Sandham (2000), Mary & Sagaut (2002) and Theofilis (2003), indicating that unsteady disturbances evolving in laminar separation bubbles may play an important role in triggering the transition process. Obviously, this raises the question of whether, and to what an extent, the associated phenomena can be captured by the theory of marginal separation, i.e. by equation (1.1). The primary motivation for the present investigation is to elucidate this question in more detail where we concentrate on near critical flows *Γ*≈*Γ*_{c}.

## 2. A bifurcation problem

As explained previously, we concentrate on the structure of solutions to equation (1.1) near the critical value of *Γ* associated with the steady two-dimensional unperturbed state, where the *A*_{∞}(0) versus *Γ* curve can be approximated by a parabola (figure 3*a*). Small derivations of *Γ* from *Γ*_{c} are then characterized by the perturbation parameter *ϵ*=(*Γ*_{c}−*Γ* )^{1/4}, and the asymptotic expansion of the wall shear in the limit *ϵ*→0 is found to be of the form . Expansions for *h* and *v*_{w} compatible with this ansatz are: , . Here, , and *A*_{∞c}(*X* ) represent the wall shear distribution in the unperturbed state with *Γ*=*Γ*_{c}, which exhibits a pronounced separation bubble as can be seen in figure 3*b* for the case of uncontrolled flow *h*_{∞}=*v*_{w∞}=0. Substitution of these expansions into (1.1) and collecting terms of *O*(*ϵ*^{2}) shows that *a*_{1} satisfies the homogeneous equation(2.1)and thus degenerates into the product of the associated right eigenfunction *b*(*X*), depending on *X* only and a shape function , which describes the variation of the wall shear disturbances with lateral distance and time: . A plot of *b*(*X*) for uncontrolled flow is included in figure 3*b*. The governing equation for *c* is derived from the solvability condition at *O*(*ϵ*^{4}) and is given by an inhomogeneous nonlinear diffusion equation of so-called Fisher type which on introducing suitably transformed quantities *u*(*z*, *t*)=(*c*/*c*_{s}+1)/2, , and can be written in parameter free form(2.2)

Here, *ν*, *μ* and *δ* are positive constants and the forcing term *g* accounts for effects which result from unsteady three-dimensional perturbations, *h*_{1} and *v*_{w1} of the steady two-dimensional obstacle shape *h*_{∞} and suction distribution *v*_{w∞} as outlined in detail in Braun & Kluwick (2004).

## 3. Finite time blow-up, before and after

It is useful to start the discussion of solutions to equation (2.2) with the most simple cases of uncontrolled two-dimensional flows where *u* is independent of *z*. Equation (2.2) then reduces to an ordinary differential equation having two stationary points *u*=(1, 0) which characterize, respectively, the upper and lower branch solutions arising under steady flow conditions. Integration with *g*=0 yields(3.1)and representative results for various initial values *u* _{0}=*u*(*t*_{0}) are depicted in figure 4*a*. It is seen that the upper branch *u*=1 is attracting (i.e. stable), whereas the lower branch *u*=0 is repelling (i.e. unstable). Furthermore, it is found that initial values *u* _{0}<0 lead to the occurrence of finite time singularities, which from a physical point of view can be interpreted as bubble bursting. Even more important, inspection of the analytical result (3.1) shows that the solutions do not terminate at the blow-up time *t**, but can be continued beyond blow-up where the flow recovers and approaches the upper branch eventually in the limit *t*→∞. In passing, we mention that an analysis similar to the one described before is possible also for above-critical flows *Γ*>*Γ*_{c}. In the case where *u* does not depend on *z*, the resulting ordinary differential equation can again be solved analytically if the forcing term is absent. In contrast to below-critical flows, the formation of finite time singularities is found to be inevitable, but as before, the solutions can be continued beyond the blow-up time. However, since a steady‐state solution does not exist any longer, a new phenomenon arises in the form of self-sustained oscillations including repeated bubble bursting with a definite frequency 1/(2*π*), figure 4*b*. Numerical solutions of equation (2.2) show that periodic bubble bursting may also occur in below-critical flows but then requires periodic forcing of sufficient amplitude.

It is interesting and important to study the blow-up properties associated with Fisher's equation (2.2) in more detail. To this end it is convenient to introduce the time shift *t*→*t*−*t** so that blow-up occurs at *t*=0. It is then easily shown that two-dimensional disturbances blow-up and recover∝1/*t* both in the below-critical and above-critical *Γ* regime.

The investigation of singularities forming in three-dimensional flow is much more complicated. Since singularities represent localized phenomena, it is natural to seek similarity solutions. Furthermore, since Fisher's equation is of a diffusive type, an obvious first guess is the ansatz , but the resulting equation for *f* has no solution. However, following lines of reasoning put forward by Hocking *et al*. (1972) in a different context, one is led to the generalization(3.2)with , as *t*→0^{∓}, which is consistent with equation (2.2). One then finds(3.3)

At this point the following observation is essential: if *u*(*z*,*t*) is large and *g*(*z*,*t*)∼O(1) so that *u*^{2}−*u*−*g*∼*u*^{2}, then equation (2.2) is satisfied also by −*u*(i*z*, −*t*). In the context of expansion (3.2), this means that the functions , after blow-up are obtained from the functions , before blow-up by replacing *η* with i*η*:(3.4)

As for two-dimensional flows, we therefore conclude that solutions of equation (2.2) may exhibit finite time singularities but can be continued beyond blow-up (figure 5*a*). According to the results (3.3), *u*(*z*, *t*) is completely smooth for *t*<0. The dip in its graph focuses continuously as *t* tends to zero and its amplitude grows as 1/*t* similar to two-dimensional flows. In striking contrast to this case, however, the flow behaviour changes drastically as *t* passes through 0. This is due to the fact that the minus signs in equation (3.4) then come into operation, which cause a singular behaviour of *u* at given by the almost parabolic solid curve in the *z*,*t* -plane. With increasing time *t*>0, therefore, *u* decreases as 1/*t* at the axis of symmetry *z*=0 but exhibits two singularities which move away from this axis with decreasing speed. By analysing the asymptotic structure (3.2) and (3.4) in more detail it may be shown that these singularities represent vortex sheets. The results depicted in figure 5*a* thus indicate that the finite time blow-up occurring in three-dimensional interactive flows governed by equation (2.2) gives birth to vortical structures moving in opposite directions. The numerical computation of the dynamics of moving singularities requires appropriate methods as, for example, that developed by Weideman (2003). Analytical continuation of the solution into the *complex* plane using Padé approximations was used to compute the path of moving complex singularities in the solution of the nonlinear heat equation *u*_{t}−*u*_{zz}−*u*^{2}=0 subjected to the initial condition *u*(*z*, *t*_{0}=−3.173 95)=cos(*z*) which appear *before* blow-up (figure 5*b*). A pair of complex conjugate singularities follow the S-shaped path until they merge at the blow-up point on the real axis. Since the local blow-up behaviour is, apart from a sign, the same as that for Fisher's equation, equations (3.2)–(3.4) strongly suggest the continuation of the solution beyond blow-up, where a pair of singularities then move along the real axis. As can be seen from figure 5*b*, the range of validity of the asymptotic expansion (3.2) extends almost up to |*t*|≈0.75.

## 4. Flow control

Practical flow conditions are always accompanied by a certain disturbance level caused by, for example, free stream turbulence. Laminar bubble bursting and the associated transition process to turbulent boundary-layer flow is therefore inevitable if the angle of attack parameter *Γ* approaches the critical value *Γ*_{c}. An effective method to increase the value of *Γ*_{c} and, consequently, to delay or even prevent the transition process near the leading edge is to place a planar surface mounted obstacle and/or a suction device into the interaction region. As an example, consider the following optimization task: let us assume a suction slot of length *L*, location *X*_{c}, the constant suction velocity amplitude *V* and the additional assumption that the suction rate , supplied, for example, by a pump, is kept fixed, see figure 6*a*. We now ask how to choose the parameters *L* and *X*_{c} to obtain the maximum value of *Γ*_{c}. Results displayed in figure 6*a* show that the curves *Γ*_{c}(*X*_{c}; *L*) exhibit a maximum for negative values of the centre location *X*_{c} of the suction device, which is pronounced for decreasing values of *L* and the associated increase of −*V*. Interestingly, the optimum is reached in the limit *L*→0(*V*→−∞). As a consequence, a narrow slot with high‐speed suction suitably placed (slightly upstream of the location of the marginal separation singularity) seems to be the most effective way to increase *Γ*_{c}. In passing, we note the relationship for the dimensional volumetric suction rate where *k*≈3.3 for the airfoil of figure 1. As pointed out in figure 6*b*, the maximum angle of attack according to classical boundary-layer theory is increased significantly by adopting an interaction strategy. As can be seen, a further remarkable increase of *α*_{m} can be achieved if boundary-layer control devices, here in the form of optimal suction investigated above, are applied.

## 5. Discussion of existing evidence and repercussions

Naturally the question arises as to whether the results of the asymptotic theory outlined so far can be confirmed by the results of numerical investigations of the full Navier–Stokes equations and experimental observations. In this connection, we refer to the direct numerical simulation study of a planar steady Blasius-type boundary-layer flow in a channel which is forced to separate marginally (Alam & Sandham 2000). Unsteady, three-dimensional perturbations (sinusoidal in lateral direction and time), in the form of a narrow suction strip placed immediately upstream of the separated region, are imposed to trigger the transition process to turbulent boundary-layer flow (figure 7). The bursting of the bubble is seen to be pronounced near mean reattachment where fluid is ejected out of the near-wall region, but nevertheless, this phenomenon remains confined to the boundary layer. Coherent structures whose shapes are similar to that of the function *b*(*X* ) (cf. figure 3*b*), arise and are washed downstream, eventually disintegrating to form the defect region of the developing turbulent boundary-layer. The almost equal spacing of the individual bursts (blow-ups or ‘spikes’) is owing to either the periodic forcing (with not necessarily the same frequency), or dominated by self-sustained oscillations if supercritical, i.e. *Γ*>*Γ*_{c} flow conditions are present. A bursting event is characterized by the generation of a *Λ*-shaped vortical structure, most conveniently displayed in the form of iso-surfaces of spanwise vorticity *ω*_{z} in the close-up of the mean separated region (figure 8). Since the leading order unsteady, three-dimensional contribution to *ω*_{z} is proportional to −*b*(*X* )*u*(*z*, *t*), the evolution of this early stage of laminar–turbulent transition is directly linked to the solution of Fisher's equation (2.2). In this connection we note that singularities predicted by Fisher's equation move in spanwise direction but are not convected downstream as in figure 8, which is expected to be an effect of higher order as far as the present analysis is concerned. Further important points to be addressed in the future include the interaction of multiple singularities that may possibly lead to chaotic behaviour and the question of whether the singularities predicted in this study can be resolved and, if so, how they affect the time-averaged flow field.

The representative snapshot of a transitional separation bubble depicted in figure 9 confirms the observation following from figure 3*b* that the values of *b* at separation and reattachment are significantly different, which means that laminar separation (1) is nearly steady, whereas reattachment is characterized by much more vigorous impact, indicated by, among other characteristics, the smoke stored in a strong recirculating eddy. Furthermore, the bulge (2) forming at the rear of the bubble reflects the shape of *b*(*X* ), signalling the onset of the bursting process accompanied by the formation of a *Λ*-vortex (3) (cf. figure 8), which eventually collapses to form the developing turbulent boundary-layer (4). An image of the time-averaged behaviour of a separation bubble on an oil coating sprayed onto an airfoil surface is given in figure 10*a*. The originally speckled structure is blurred through the acting of the wall shear, except for the range within the bubble, where the shear is too low to streak the oil. The reattachment region is characterized by the action of highly unsteady shear and the appearance of a dividing line (the mean reattachment line). Upstream of this line, the oil is driven contrary to the main flow direction and forms the white ‘oil accumulation line’. The sketch of the corresponding mean wall shear distribution (suggested, for example, by the calculations of Alam & Sandham 2000), depicted in figure 10*b*, shows the pronounced minimum of the mean wall shear near reattachment and the associated recirculation eddy.

In a recent publication, Borodulin *et al*. (2002) have argued that bursting processes in boundary-layer flows share common properties that are not dependent on the specific problem under consideration. This suggests that an analysis similar to the one carried out here should be possible also in the context of classical triple deck theory, where a number of problems are known to exhibit non-unique solutions. This is supported by recent work in progress.

## Footnotes

One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.

- © 2005 The Royal Society