## Abstract

Suction into small two- or three-dimensional surface slots inside an otherwise planar boundary-layer is examined theoretically through a combined analytical and computational approach. Increasing suction strength leads to enhanced nonlinear structures with flow reversals or trailing vortices.

## 1. Introduction

The effects of slot suction in an otherwise solid surface supporting a boundary layer are important in laminar flow control (Joslin 1998; Ovenden 2001) where, typically, an array of slots is deployed (Ellis & Poll 1996). The current interest arose from industrial concerns about the creation of vortices and similar complexities by suction slots buried well within the boundary layer. Consequently, the particular focus here is on understanding the nonlinear steady influence from a small isolated slot, as a precursor to the case of an array, for an incompressible laminar flow.

Both two-dimensional and three-dimensional models merit study. The former allows far more progress in following the effects of increased suction velocities and, hence, nonlinearity, whereas three-dimensional flow studies reveal the subtleties in the wake of the slot. Some of the ideas and scales from two-dimensional cases apply equally well in three dimensions, while others do not, according to the present theory on the flow structure accompanied by analytical and computational results for high Reynolds numbers. Specifically, the three-dimensional setting is partly with a view to obtaining insight into the generation of streamwise vortices, for example, horseshoe vortices trailing downstream of a slot (Ovenden & Smith 2003). The two-dimensional setting, however, is concerned with spanwise vortices or, more commonly, flow reversals created over and beyond the slot. Issues include the first appearance of longitudinal vortices, their form, and the appearance of flow reversal as suction amplitudes increase. Again, both two-dimensional and three-dimensional settings are found to suggest over-suction (Ellis & Poll 1996; Joslin 1998), which remains a practical limitation on the usefulness of suction in laminar flow control.

In this paper, we consider an incident two-dimensional boundary-layer encountering a two-dimensional or three-dimensional slot with some prescribed suction–velocity distribution, in contrast with the investigation of Smith *et al*. (2003) of flow inside a slot. The flow is taken to be incompressible, steady and is described using Cartesian coordinates , and representing the streamwise, normal and spanwise directions, respectively, with corresponding velocities , and . The pressure is given by and the variables are taken to be non-dimensional, with large Reynolds number. In §§2 and 3, we address the two-dimensional and three-dimensional cases in turn, where the slot length is assumed to be comparable to, or smaller than, the local boundary-layer thickness. Further comments are provided in §4.

## 2. Two-dimensional slots

When the two-dimensional suction slot length is some factor *δ*≪1 of the local unperturbed boundary-layer thickness and the suction flow rate is not too strong, the dominant upstream influence is limited to the slot-length scale only (cf. longer slots in Ovenden 2001) in a manner similar to that seen in Smith (1978) and Smith & Walton (1998). The main outer region is of length and height dimensions comparable to the slot length, implying that with *x*, *y* of O(1). The suction slot, therefore, only interacts significantly with the near-wall shear flow of the upstream velocity profile (figure 1*a*), in the form(2.1)Here, *λ*∼1 is the normalized incident shear and *μ* measures the suction velocity, such that the slot has mass flux O(*μδ* Re^{−1/2}). Provided that *μ*≪*δ* and , the local region is inviscid and *v*(*x*, *y*) satisfies Laplace's equation, ∇^{2}*v*=0, subject to *v* tending to zero in the far-field while *v*=*V*_{w}(*x*) (the suction profile) is imposed at *y*=0. The solution is(2.2)where *u*_{slip}(*x*) is the streamwise-induced slip velocity at the surface. For a localized suction profile applied in the vicinity of *x*=0,(2.3)implying that *u*_{slip}(*x*) reverses sign and decays as *x*^{−1} in the far-field.

Various suction amplitudes *μ* can now be considered. As *μ* is increased, the subtle changes in structure and physics are caused primarily by a thinning of the viscous region over the slot. This, in turn, enables the elliptic flow response from the upper inviscid layer to interact strongly with the flow in the close neighbourhood of the slot, while remaining negligible elsewhere. In all cases examined, the applied suction profile is assumed to be non-zero only between *x*=0 and *x*=1 and to tend to zero linearly at these points, avoiding any strong nonlinear behaviour locally. Here, we concentrate on a single strong suction case (see Ovenden 2001 for other weaker and intermediate suction cases).

When *μ*=δ^{1/3}Re^{−1/6}, the entire viscous response is nonlinear because the slip velocity generated by (2.2) is comparable to the leading-order streamwise shear flow in the viscous sublayer. In the sublayer, , with *Y*∼1 and(2.4)The above expansion leads to six distinct regions of flow neighbouring the slot, the locations of which are shown in figure 1*a*.

Owing to the strong suction, the flow in region 1 (figure 1*a*) over the slot where *V*_{w}(*x*)≠0 takes a simple shear form, such that and *U*(*x*,*Y*)=*Y*+*u*_{slip}(*x*). Beneath this region, a thinner layer (region 2) exists to satisfy the no-slip condition at the slot surface. The solution there is given by(2.5)where . Flow reversal occurs over the suction slot where *u*_{slip} changes sign from positive to negative. For a symmetrical suction profile, this is at the midpoint of the slot.

Upstream of the slot in region 3, *V*_{w}(*x*)=*p*_{0}(*x*)=0 and the boundary-layer equations(2.6)hold. The boundary conditions of no-slip, matching to the outer inviscid region and an initial condition upstream, then require(2.7)The upstream influence of the suction slot is felt in region 3 through the induced slip velocity in (2.7). The system consisting of (2.6) and (2.7) was solved numerically. The results for a typical symmetrical suction profile indicate a strong favourable pressure gradient accelerating the flow towards the slot's upstream edge. A scaling argument reveals that the pressure drop over the upstream wall increases proportionally to the suction strength squared, max|*V*_{w}(*x*)|^{2}. Plots of the upstream skin friction for various suction strengths are presented in figure 1*c*.

At the upstream edge, the flow yields a strongly attached profile, which must be altered rapidly within some thin edge region (region 4) to match with the shear profile obtained above the slot in region 1. Here, region 4 has and and, since(2.8)these produce the thin-layer Euler equations(2.9)governing the leading-order solution. This is subject to the arbitrary edge profile *U*=*U*_{0}(*Y*) and pressure at *X*=0, an outer matching condition given by *U*∼*Y*+*u*_{slip}(0) as *Y*→∞, and the near-edge suction profile *v*=−*ΓX* at *Y*=0 for *X*>0 (*Γ*=−*V*′_{w}(0)>0). The solution was obtained by a variation of an analysis used by Messiter *et al*. (1973) and Bowles & Smith (2000), where the coordinate system is changed from (*X*,*Y*) to (*X*,*Ψ*) and *Ψ* is the stream function . As a result, we obtain the following integral equation for ,(2.10)Solving this equation confirms that the flow tends to a simple shear form *U*=*Y*+*U*_{slip}(0) over the hole (see figure 1*b*). The excess vorticity generated by the flow passing over the upstream wall is sucked down into the suction slot and lost within an asymptotically finite distance of the upstream edge.

Downstream of the midpoint of a symmetrical suction slot (more generally the point where *u*_{slip}(*x*)=0), a region of reversed flow appears (shaded in figure 1*a*), which grows in height towards the downstream edge. The maximum height of this region is directly proportional to the suction strength, and is attained just ahead of the slot's downstream edge.

The governing equations beyond the downstream edge, in region 6, are the same as in (2.6) but with *x*>1, (*U*,*V*)=(0,0) at *Y*=0 and *U*∼*Y*+*u*_{slip}(*x*) as *Y*→∞; in this case, *u*_{slip}<0. The forward part (*U*>0) of the initial velocity profile at *x*=1 is obtained from the thin Euler region of rapid adjustment (region 5) that is equal in size to region 4 and situated over the downstream edge; the reversed part of the profile remains unknown. The conservation of on streamlines in region 5 indicates that the forward part of the flow must take the form of a straight line, fixed by the outer boundary condition. Results shown in figure 1*d–f* for a typical symmetrical suction profile are for three different suction strengths, max|*V*_{w}(*x*)|. Note that reattachment occurs fairly close to the downstream edge owing to *u*_{slip}→0 far downstream. A very strong adverse pressure gradient remains localized near the downstream edge, accelerating the reversed flow back into the slot. This explains the large negative skin-friction values at the downstream edge shown in figure 1*d*, leading to strongly reversed (but attached) velocity profiles at the downstream edge. Streamline plots for the cases of max|*V*_{w}(*x*)|=2 and 3 are given in figure 1*e*,*f*, respectively. Can we understand how this region of reversed flow grows as the suction strength increases further? From the definition of *u*_{slip}(*x*) in (2.2), it is straightforward to conclude that over the downstream edge the induced slip velocity rapidly assumes the asymptotic form , where *α* represents the suction strength. Rescaling the governing equations (2.6) and boundary conditions given above for *α*≫1, we find that , indicating that the length of the reversed flow region grows downstream as the three-quarter power of the suction strength.

## 3. Three-dimensional slots

The problem set-up is similar to that in §2 but is now three-dimensional with an isolated suction hole of radius O(*δ* Re^{−1/2}), and the suction strength parameter *μ*=*δ*^{−1/3}Re^{−1/3} is much weaker than in two dimensions. Therefore, we have(3.1)and the three-dimensional nonlinear viscous layer equations with unknown pressure *p*(*x*,*z*),(3.2)(3.3)apply. The conditions of normal wall suction and far-field decay still hold, but now with zero displacement, thus(3.4)We note that the two-dimensional solution to this problem for a slot where *V*_{w}(*x*,*z*)=*V*_{w}(*x*) simply has , allowing the progress to higher suction rates such as in §2. If |*V*_{w}|≪1, then a linearized three-dimensional solution is valid, which can be obtained via a double Fourier transform defined for an arbitrary function *g*(*x*,*y*,*z*) as . The solution obtained gives *v*≡*V*_{w}(*x*,*z*) throughout, ,(3.5)where , are the streamwise and spanwise skin frictions, respectively, and *d*(*x*,*z*) describes the spanwise flow behaviour far from the surface. The linearized solution predicts completely planar flow (*v*=0) at all points not directly situated above the hole itself.

If |*V*_{w}|∼1, then a numerical scheme is generally needed, which must contend with the added ellipticity of the pressure field in three dimensions. The method chosen and modified is the so-called ‘skewed shears’ method developed in Smith (1983) and is based on computing (∂_{x}*u*+∂_{z}*w*) and ∂_{x}*v* as a set of parallel, two-dimensional parabolic problems with the Poisson equation for the pressure treated separately at each iteration. The scheme was tested by reproducing the linear result for small |*V*_{w}|. The computation was then run with a suction profile of the form for (*x*^{2}+*z*^{2})<1 and zero otherwise. The parameter represents the suction strength and various results are displayed in figure 2.

Figure 2*a*,*b* shows the streamwise skin friction perturbation for both linearized *C*≪1 (evaluated numerically) and nonlinear *C*=1 cases, respectively. In both cases, a corridor downstream of the slot can clearly be seen containing fluid that has been accelerated by the suction hole but has subsequently overshot its downstream edge. On increasing the suction strength from linear to nonlinear, a fork-shaped pattern moves upstream from infinity and the corridor broadens slightly. The stronger suction thus appears to decelerate the corridor flow on the symmetry plane *z*=0 while accelerating the fluid located on either side. The upstream movement of such nonlinear effects in the corridor as the suction strengthens suggests that the linear expansion of the solution eventually becomes invalid when *x*≫1 and *z*∼1 owing to an overtake by higher-order terms at low suction rates. Outside the corridor, linear and nonlinear flow fields are indistinguishable from one another, and *τ*_{x} attains a sharp minimum at the spanwise extremities of the suction hole.

Other evidence for the fully nonlinear corridor is provided by comparing the plots of *d*(*x*,*z*), describing the spanwise flow far from the surface, for linear and nonlinear cases in figure 2*c*,*d*, respectively. The comparison demonstrates that the symmetry of *d*(*x*,*z*) upstream and downstream of the hole predicted in the linear case is disrupted by stronger suction. Hence, it appears that, for *C*∼1, not all of the fluid far from the surface is drawn towards the plane *z*=0, but that a severe deceleration within the corridor pushes more fluid outwards to satisfy mass conservation. No change in spanwise behaviour is observable outside the corridor.

Combining the observations above with the additional information from the spanwise skin friction *τ*_{z} in figure 2*e*, we can build up the following picture of the entire flow field. The near-wall flow is pulled strongly inwards from both sides upstream of the hole centre, accelerating the near-wall streamwise flow. However, on approaching the downstream edge of the hole, the flow stops accelerating streamwise and *τ*_{x} attains its maximum. In response, a spanwise flow reversal takes place, so that fluid close to the surface begins to be pushed outwards away from the symmetry plane *z*=0 while the flow above is still being pulled inwards.

In the linear regime, *no* vortices are present downstream as a result of the negligible *v* component. This is not the case for stronger suction and an examination of the vertical velocity field suggests that the nonlinear wake consists of a three-column structure extending downstream of the hole. The central column spanning the symmetry plane *z*=0 gives rise to a spanwise jet-like structure where all the fluid is pushed outwards and upwards from the *x*-axis by a severe deceleration in the streamwise direction. This fluid then feeds a pair of near-wall vortices either side, the centre-lines of which are roughly parallel to the *x*-axis and extend downstream from the spanwise extremities of the suction hole. Comparing the magnitudes of *v* and *w* in the wake indicates that the near-wall vortices are elongated spanwise. Figure 2*f* displays a sketch of the secondary flow field in the hole wake. The fact that these near-wall vortices appear to remain spaced apart by the width across the hole was confirmed by repeating the computations with a suction hole elongated in the spanwise direction.

Concerning even stronger suction, figure 3 presents details of the flow field surrounding a very strong suction hole with *C*=6.0. Examining initially *τ*_{x} in figure 3*a*, we see that at higher suction strengths, the fork-shaped pattern becomes increasingly stretched as the velocity difference between the symmetry-plane flow and the flow either side grows sharply. In fact, the deceleration along the centre-line is so intense in this case that *τ*_{x} drops below its unperturbed value around *x*=5.0 and appears to keep falling further downstream. This indicates the possibility of flow separation at higher suction rates, and offers a potential link with the two-dimensional case. Spanwise velocity information from figure 3*b*,*c* indicates that an increasing amount of fluid is drawn in spanwise by the strong suction. Downstream of the hole, within the corridor, the region of flow outwards from the symmetry line far from the surface moves further upstream towards the hole as expected. The original pair of trailing vortices thus strengthens and moves further apart under increasing suction. However, more surprisingly, in the spanwise skin friction in figure 3*c*, the strong suction appears to have produced a further change of sign close to the symmetry line at *x*≈4. Given that the vertical velocity maintains the same structure as that noted in the *C*=1 case, this new feature suggests that two new counter-rotating vortices are created inbetween the outer vortex pair. Briefly comparing the vertical and spanwise velocities further indicates that these new vortices exist and are elongated normal to the surface. A sketch of the secondary flow for this case is included in figure 3*d*.

## 4. Further comments

We have shown that strong suction through short two-dimensional slots acting on a boundary layer accelerates the oncoming flow upstream, but creates large-scale flow reversal on the downstream side; this is not ideal for maintaining laminar flow. In three-dimensional cases, the numerical computations have given us some idea of the processes of over-suction (perhaps via the production of unstable growing vortices), but investigations of even stronger suction appear necessary to understand over-suction fully. An interesting question is how the flow will adapt under further increases in suction through an isolated hole. The suction could pull more fluid in from the sides increasing the strength of the trailing vortices or, alternatively, the hole could drag fluid close to the centre-line back upstream, creating three-dimensional flow separation. The behaviour of such strong suction holes in an array also merits similar investigation.

Comparisons have been made with two-dimensional numerical simulations of suction slots by Dr Christopher Davies (2001, personal communication) and these are found to predict downstream flow reversal for short slots. In three dimensions, the appearance of an extra pair of vortices at higher suction rates agrees with the numerical and experimental findings of MacManus & Eaton (2000).

## Acknowledgments

We thank EPSRC and QinetiQ (formally DERA) for their financial support and Dr Chris Atkin for his help and encouragement throughout. We are also grateful to Dr Christopher Davies for his interest and numerical comparison of the two-dimensional case, and to the referees for their encouraging comments and suggestions.

## Footnotes

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

- © 2005 The Royal Society